D ' Abord

Je Voudrais Remercier

François Mr

Jouve

Philippe Mr

Helluy

Je Remercie Également Mme

Aguillon Nina

Mr Mme Debit Naima

Mr Fadili Jalal

Forcadel Nicolas D'

Je Voudrais

Comba, Etienne, Tuyen, Tin, Angelot, Julien, Vlere Arnaud

Leonard

Hong

Stavroula

Dalal

Thuen Ruben Guillaume

Youssef, Khalid, Toufik, Soufiane, Said, Mounir Ali

également reconnaissant pour le temps conséquent qu'il m'a accordé, ses qualités pédagogiques et scientifiques, sa franchise et sa sympathie. J'ai beaucoup appris à ses côtés, en particulier sur le plan théorique et numérique et je lui adresse ma gratitude pour le temps consacré avec ses conseils, son implication et les corrections apportées à cette thèse.

À l'âme de ma Mère

Introduction

Un fluide est un milieu matériel parfaitement déformable sous l'action d'une force extérieure. En général on a deux types de fluides, fluides compressibles tels que les gaz et plasma et des fluides incompressibles tels que les liquides. La trajectoire d'une particule décrit le mouvement selon l'écoulement du fluide laminaire ou turbulent. Le premier type d'écoulement est caractérisé par des lignes de courant parallèles qui se traduisent par un faible nombre de Reynolds contrairement au deuxième type d'écoulement où le nombre de Reynolds est élevé et qui présente en tout point un caractère tourbillonnaire. Dans le cadre de cette thèse, nous nous intéressons à l'étude théorique et numérique de ces deux types de modèles d'écoulement, en particulier les écoulements dans des milieux poreux.

Pour faire face à chaque type d'écoulement et sa mise en oeuvre, nous nous intéressons à l'optimisation des structures hydrauliques telles que les passes à poissons. Ces structures jouent un rôle d'un pont de passage entre le milieu marin et les rivières favorisant et facilitant la migration quotidienne ou annuelle des espèces aquatiques des eaux salées vers les eaux douces pour la reproduction et pour accomplir leurs cycles biologiques. A titre d'exemple largement connu, le saumon qui est capable de nager des centaines de kilomètres en amont. Quand une barrière artificielle est construite dans un ruisseau, des règlements juridiques européens obligent l'installation d'une passe migratoire afin de permettre aux espèces de surmonter la barrière. Les passes à poissons sont des structures hydrauliques placées à coté des barrages permettant aux poissons de traverser vers l'amont du fleuve. La plupart de ces structures sont construites en pente divisées en plusieurs bassins. Dans la littérature, de nombreux types de passes à poissons existent, celles de Denil [START_REF] Katopodis | Denil Fishways of varying[END_REF], ou bien à fentes verticales VSF (Vertical Slot Fishway) [START_REF] Rajaratnam | Hydraulics of vertical slot fishways[END_REF]. Ce dernier type fait l'objet de notre étude. Du point vue physique, la vitesse de l'écoulement du fluide dans ces structures est relativement faible permettant des pauses de repos. De plus ces structures permettent une bonne dissipation d'énergie. Ces caractéristiques nous ont conduit à privilégier et optimiser le choix de ce type de passe à poissons.

Un problème d'optimisation de forme consiste à trouver une forme dite optimale dans un ensemble de domaines admissibles qui minimise une fonction objective reflétant les conditions de confort pour les poissons. La recherche liée à l'optimisation de forme fait partie du domaine de contrôle optimal gouverné par des équations aux dérivées partielles. Le problème de contrôle intervient dans plusieurs domaines d'applications tels que l'ingénierie, la construction navale, aéronautique, génie civil et autres. Au fil de ces dernières années, cet horizon de recherche attire de plus en plus l'attention la communauté mathématiques appliquées et ingénierie [START_REF] Lions | Optimal control of systems governed by partial differential equations[END_REF][START_REF] Gunzburger | Perspectives in Flow Control and Optimization[END_REF][START_REF] Wang | Aerodynamic optimization design of centrifugal compressor's impeller with Kriging model[END_REF][START_REF] Mohammadi | Applied Shape optimization for fluids[END_REF][START_REF] Yan | Shape inverse problem for the two-dimensional unsteady Stokes flow[END_REF][START_REF] Maury | Elasto-plastic Shape Optimization Using the Level Set Method[END_REF]. Pour répondre à notre problématique, nous explorons différents types de modèles d'écoulements. Dans un premier lieu, nous introduisons un système hyperbolique des eaux peu profondes. Ce modèle dérivant des équations de Navier-Stokes incompressible en utilisant la méthode d'approximation de Boussinesq où les variations de densité sont négligeables sauf l'effet gravitationnel. Ce modèle est largement utilisé dans le cas de la modélisation des phénomènes géophysiques à surface libre pour simuler de nombreux phénomènes naturels en raison de son efficacité. Le but de l'étude est de trouver une structure optimale de la structure hydraulique permettant aux poissons de franchir l'obstacle dans des conditions convenables. Nous donnons une formulation mathématique du canal composé de dix bassins avec des fentes verticales afin d'obtenir un débit moyen adapté à une grande variété d'espèces migratrices. Les équations bidimensionnelles en eaux peu profondes avec terme source s'écrivent

∂H ∂t + ∇. Q = 0 dans Ω × (0, T ) ∂ Q ∂t + ∇.( Q ⊗ Q H ) + gH ∇(H -η) = f dans Ω × (0, T ) (0.0.1)
avec H est la hauteur de l'eau; -→ u = (u 1 , u 2 ) est le vecteur de vitesse; u 1 et u 2 sont les composantes x et y de la vitesse d'écoulement, respectivement; Q = (u 1 H, u 2 H) est le débit d'eau; η est la géométrie du fond; g est l'accélération de gravité et -→ f représente tous les effets de frottements du fond et les forces de la pression atmosphérique. Nous associons au système de Saint-Venant bidimensionnel les conditions initiales et aux limites suivantes

H(0) = H 0 , Q(0) = Q 0 dans Ω Q. n = 0, curl Q H = 0 sur γ 0 × (0, T ) Q = Q 1 n sur γ 1 × (0, T ) H = H 2 sur γ 2 × (0, T ) (0.0.2)
avec γ 0 est le bord latéral, γ 1 et γ 2 sont les bord d'entrée et de sortie respectivement. L'optimisation de forme est liée à une fonction objective qui dépend des capacités de nage du poisson. Tout d'abord, nous calculons sa dérivée de forme en fonction des variables d'état et des variables adjointes en dérivant un système adjoint. Les résultats d'existence et d'unicité du système adjoint sont prouvés à l'aide des techniques pseudo-différentielles. Une méthode de volume fini est adoptée pour résoudre le problème d'état (0.0.1) combinée avec deux algorithmes de minimisation (algorithme de simplexe et gradient projeté) pour la résolution du problème d'optimisation de forme.

Le deuxième type de modèle basé sur un écoulement peu profond est un modèle multicouches. Le modèle multicouches est introduit pour éviter les calculs coûteux des équations de Navier-Stokes lorsque le débit est encore dans le régime peu profond.

Commençons par les équations de Navier-Stokes non stationnaires

∂ t ρ + ∇.(ρ u) = 0 dans Ω × [0, T ] ∂ t (ρ u) + ∇.(ρ u ⊗ u) = ∇.Σ T + ρg dans Ω × [0, T ] (0.0.3)
avec Σ T est le tenseur donné par

Σ T = -pId + µ(∇ u + (∇ u) ′ )
et g = (0, -g) ′ ∈ R d est l'accélération de gravité, et µ est une viscosité donnée.

En suivant la méthode initiée par Audusse et al. [START_REF] Audusse | Numerical simulations of 3D free surface flows by a multilayer Saint-Venant model[END_REF] . La hauteur de l'eau est divisée en un certain nombre de couches où l'hypothèse de Saint-Venant peut être appliquée à chaque couche. En considérant une vitesse verticale non constante, le profil vertical de la vitesse horizontale est décrit par le système multicouches suivant

∂ t h α + ∇ x .(h α u H,α ) = G α+ 1 2 -G α-1 2 . ρ∂ t (h α u H,α ) + ρ∇ x .(h α u H,α ⊗ u H,α ) -∇ x . µh α (∇ x u H,α + (∇ x u H,α ) ′ ) = 1 2 ρG α+ 1 2 ( u H,α+1 + u H,α ) -1 2 ρG α-1 2 ( u H,α + u H,α-1 ) -µ( K α+ 1 2 -K α-1 2 ) -ρgh α ∇ x (z B + h) -h α ∇ x p S .
(0.0.4) avec G α+1/2 est le flux normal de masse à l'interface entre les couches α -1 et α.

Nous montrons l'existence et l'unicité d'une solution forte du système multicouche bidimensionnel. Cela se fait en plusieurs étapes: l'analyse du problème linéarisé, l'obtention d'une estimation à priori et enfin l'utilisation d'un schéma de Picard pour le traitement du cas non linéaire. Afin d'étudier le problème d'optimisation de forme, nous procédons à la généralisation de la technique multicouches aux équations hydrostatiques de Navier-Stokes tridimensionnelles et dérivons un système multicouches 3-D. La description de l'écoulement multicouches de Saint-Venant permet de déduire une structure optimale de la passe à poissons en minimisant une fonction coût donnée modélisant la facilité de passage du poisson à travers la structure vers les zones de reproduction, de croissance ou d'alimentation. Nous dérivons un système adjoint lié au modèle multicouche 3-D obtenu et exprimons la dérivée de domaine de la fonction objective en termes des variables d'état et des variables adjointes. Nous présentons deux procédures de minimisation: la première est un algorithme de simplexe et la deuxième est un algorithme de type gradient (gradient projeté).

Le troisième type d'écoulement d'eau est basé sur le modèle de milieux poreux stationnaire. L'écoulement des fluides dans les milieux poreux est généralement décrit par les équations de Brinkman Forchheimer [START_REF] Forchheimer | Wasserbewegung durch boden[END_REF]:

-ν∆u + (u.∇)u + a | u | α u + ∇p = f in Ω; ∇.u = 0 dans Ω u = 0 sur Γ 0 u = g sur Γ 1
ν ∂u ∂npn = 0 sur Γ 2 (0.0.5) avec u et p représentent respectivement la vitesse et la pression. ν est le coefficient de Brinkman, a est le coefficient de Forchheimer. f désigne les forces extérieures et α ∈ [START_REF] Clay | Design of fishways and other Fish Facilities[END_REF][START_REF] Katopodis | Denil Fishways of varying[END_REF] est un nombre réel choisi de manière appropriée tout au long de ce travail. Γ 0 est le bord latéral, Γ 1 est le bord d'entrée, du haut et du fond, et Γ 2 est le bord de sortie. Dans un premier temps, nous avons examiné si le problème est bien posé et nous avons montré l'existence et l'unicité d'une solution faible du problème (0.0.5).

Parmi les difficultés liées à l'approximation numérique de la solution du problème (0.0.5), la contrainte d'incompressibilité "div u =0" puisque le champs de vitesse u et la pression son dissociés. De multiple méthodes sont utilisées pour la relaxation de cette contrainte. Nous pouvons citer:

1. La méthode de pénalisation. Nous pouvons citer comme référence; J. Shen [START_REF] Shen | On error estimates of the penalty method for unsteady Navier Stokes equations[END_REF], et R. Temam [START_REF] Temam | Une méthode d'approximation de la solution des équations de Navier-Stockes[END_REF] ∇.u ε + εp ε = 0 in Ω × [0, T ] 2. Compressibilité artificielle, nous rappelons les travaux de Chorin [START_REF] Joel | Numerical solution for the Navier Stokes equations[END_REF] et de R. Temam [START_REF] Temam | Sur l'approximation de la solution des équations de Navier Stokes par la méthode des pas fractionnaires (ii)[END_REF] ∇.u ε + εp ε t = 0 in Ω × [0, T ]

3. La pression stabilisée

∇.u ε -ε∆p ε = 0 in Ω × [0, T ], ∂p ε ∂n = 0 in ∂Ω × [0, T ]
introduit par Temam [START_REF] Temam | Une méthode d'approximation de la solution des équations de Navier-Stockes[END_REF] et Chorin [START_REF] Joel | Numerical solution for the Navier Stokes equations[END_REF][START_REF] Joel | On the convergence of discrete approximations to the Navier Stodes equations[END_REF]. 4. La méthode de pseudo-compressibilité

∇.u ε -ε∆p ε t = 0 in Ω × [0, T ], ∂p ε ∂n = 0 in ∂Ω × [0, T ]
Durant ce travail, une méthode de pénalisation est utilisée pour relaxer la contrainte d'incompressibilité de la vitesse. Le problème perturbé associé au problème (0.0.5) s'écrit

-ν∆u ε + (u ε .∇)u ε + 1 2 (∇.u ε )u ε + a | u ε | α u ε + ∇p ε = f dans Ω ∇.u ε + εp ε = 0 dans Ω u ε = 0 sur Γ 0 u ε = g sur Γ 1 ν ∂u ε ∂n -p ε n = 0 sur Γ 2 (0.0.6)
Afin d'évaluer l'utilité de l'approche de pénalisation, nous dérivons des estimations à priori utiles pour entamer l'analyse de l'erreur. Nous utilisons une méthode de Faédo-Galerkin pour prouver l'existence d'une solution faible du problème perturbé (1.0.6). Le problème d'optimisation consiste à minimiser une fonction coût. Nous exprimons son gradient de forme en fonction de la vitesse u comme variable d'état, des variables adjointes, et le vecteur unité normal au bord du domaine. Ces dernières années, la méthode d'éléments finis est utilisée dans plusieurs domaines d'ingénierie et d'analyses numériques [START_REF] Ciarlet | The Finite Element Method for Elliptic Problems[END_REF][START_REF] Schneider | Highly gravity-driven flow of a NAPL in water-saturated porous media using the discontinuous Galerkin finite-element method with a generalised Godunov scheme[END_REF][START_REF] Zienkiewicz | Finite Element Method in Engineering Science[END_REF]. Nous adoptons une méthode d'éléments finis discrète pour approximer la solution du problème pénalisé et établir des estimations à priori afin de prouver la convergence de la solution approchée vers la solution de l'équation de Navier Stokes Forchheimer. Le problème d'optimisation est implémenté en utilisant la méthode adjointe continue et la méthode d'éléments finis.

Le dernier type de modèle d'écoulement est décrit par les équations de Navier-Stokes Forchheimer non stationnaires

u t -ν∆u + (u.∇)u + a | u | α u + ∇p = f dans Ω ∇.u = 0 dans Ω u = 0 sur Γ 0 u = g sur Γ 1
ν ∂u ∂npn = 0 sur Γ 2 u| t=0 = u 0 (0.0.7)

Dans un premier lieu, nous montrons l'existence et l'unicité d'une solution faible du problème (0.0.7). Les estimations d'erreur de la vitesse et de la pression sont établies à l'aide d'une méthode d'énergie.

Ensuite, nous entamons l'étude du problème perturbé en deux parties; la première est consacrée au traitement du problème linéarisé. La deuxième partie traite le problème non linéaire.

Le problème perturbé prend la forme

u ε t -ν∆u ε + (u ε .∇)u ε + 1 2 (∇.u ε )u ε + a | u ε | α u ε + ∇p ε = f dans Ω ∇.u ε + εp ε = 0 dans Ω u ε = 0 sur Γ 0 u ε = g sur Γ 1 ν ∂u ε ∂n -p ε n = 0 sur Γ 2 u ε | t=0 = u 0 (0.0.8)
Une discrétisation en temps est mise en place en utilisant un schéma d'Euler. Ce dernier est combiné à une discrétisation totale d'éléments finis discrète pour approcher le problème perturbé et dériver une estimation d'erreur de la vitesse et de la pression afin de montrer la convergence de la solution approchée vers la solution du problème initial (0.0.7).

Le problème d'optimisation de forme consiste à trouver une forme optimale de la structure en minimisant une fonction coût. Nous établissons un système adjoint associé au problème perturbé puis nous calculons le gradient de forme de la fonction à minimiser en fonction des variables d'état et adjointes. Le problème d'optimisation est implémenté en utilisant la méthode adjointe continue et la méthode d'éléments finis.

Dan ce qui suit, nous détaillons le lien entre les différents types de modèles d'écoulements mentionnés et les chapitres de cette thèse. Dans le premier chapitre, nous considérons les équations des eaux peu profondes bidimensionnelles (0.0.1) avec les conditions initiales et aux bords (0.0.2). Nous traitons ces équations en cinq sections. La première section est consacrée à la formulation mathématique de la structure du VSF. Nous donne la définition de la fonction coût à minimiser comme suit

J = 1 2 T 0 Ω - → u -- → v 2 dxdt + α 2 T 0 Ω | rot( - → u ) | 2 dxdt (0.0.9) avec - → u = - → Q H
est la vitesse de l'eau et (H, -→ Q ) sont solutions du système (0.0.1)-(0.0.2). La vitesse -→ v est une vitesse cible liée à la capacité de nage des poissons. Elle s'écrit -→ v (x 1 , x 2 ) = (c, 0) si x 2 ≤ 1 3 0.97 (0, 0) sinon (0.0.10) La deuxième section est concernée au calcul du système adjoint associé aux équations (0.0.1) et la fonction coût (0.0.9). Le système adjoint obtenu prend la forme

- ∂p ∂t + 1 H 2 ( Q. ∇) r. Q -gH( ∇. r) -g ∇η. r = -( Q H -v) Q H 2 -α curl(curl( Q H )). Q H 2 - ∂ r ∂t -∇p - 1 H ( Q. ∇) r - 1 H ( ∇ r) t Q = 1 H ( Q H -v) + α H curl(curl( Q H
)) (0.0.11) avec les conditions aux limites et finale suivantes p(T ) = 0, r(T ) = 0 dans Ω r. n = 0 sur γ 0 × (0, T )

(gH - Q 2 1
H 2 ) r. n = 0 sur γ 1 × (0, T )

{p + 1 H 2 ( Q. r)} n + 1 H 2 ( Q. n) r - α H 2 curl( Q H 2 ) τ = 0 sur γ 2 × (0, T ) (0.0.12)
avec (H, Q) ∈ L 2 (0, T ; R) × H 2 (0, T ; Ω) est solution du système (0.0.1), et (p, r) ∈ (L ∞ ([0, T ]; H 2 (Ω))) 2 . la troisième section est dédiée à la dérivation de domaine de la fonction coût (0.0.9) en fonction des variables d'état (H, Q) les variables adjoints (p, r).

Théorème 0.0.1. Soit V ∈ Lip(Ω; R 2 ). Si les fonctions A, J et L sont différentiables par rapport au domaine Ω dans un ensemble admissible X 0 , alors la dérivée du domaine de la fonction j à Ω ∈ X 0 prend la forme ∂ ∂Ω j(Ω). V = ∂ ∂Ω J(Ω; H, Q). V -∂ ∂Ω A(Ω; H, Q; p, r). V + ∂ ∂Ω L(Ω; p, r). V (0.0.13)

avec ∂ ∂Ω L(Ω; p, r). V = T 0 Ω f . r( ∇. V ), f or V ∈ Lip(Ω; R 2 ) ∂ ∂Ω J(Ω; H, Q).. V = 1 2 T 0 Ω || Q H -v|| 2 ( ∇. V ) + 1 2 T 0 Ω |curl( Q H )| 2 ( ∇. V ) ∂ ∂Ω A(Ω; H, Q; p, r). V = T 0 Ω ∂H ∂t p( ∇. V ) + T 0 Ω ( ∇. Q)p( ∇. V ) - T 0 Ω ( ∇ V ) t : ∇ Qp + T 0 Ω ∂ Q ∂t . r( ∇. V ) + T 0 Ω ( ∇. Q) Q H . r( ∇. V ) - T 0 Ω ( ∇ V ) t : ∇ Q Q H . r + T 0 Ω ( Q. ∇) Q H . r( ∇. V ) - T 0 Ω ( ∇ V Q. ∇) Q H . r + T 0 Ω gH ∇H. r( ∇. V ) T 0 Ω gH( ∇ V ) t ∇H. r - T 0 Ω gH ∇η. r( ∇. V ), f or V ∈ Lip(Ω; R 2 )
pour V ∈ Lip(Ω; R 2 ), et (H, Q) est solution du système (0.0.1) avec les condition initales et aux bords (0.0.2) et (p, r) est solution du système adjoint (0.0.11) avec les conditions finales et aux bords (0.0.12).

La quatrième section regroupe des résultats concernant l'existence et l'unicité du système adjoint (0.0.11). Nous commençons par l'écrire sous une forme conservative

∂ W ∂t + M (H, Q) ∂ W ∂x + N (H, Q) ∂ W ∂y = S(H, Q, W ) (0.0.14) avec W = (p, r 1 , r 2 ) t , c = gH M (H, Q) =    0 c 2 -u 2 1 -u 1 u 2 1 2u 1 u 2 0 0 u 1    , S(H, Q, W ) =    F -g ∇η. r -G 1 -G 2    et N (H, Q) =    0 -u 1 u 2 c 2 -u 2 2 0 u 2 0 1 u 1 2u 2   
Ensuite, nous montrons l'existence et l'unicité de la solution du système (0.0.14) en utilisant les opérateurs pseudo-différentiels Théorème 0.0.2. Si W 0 ∈ H s (R 2 ) et S ∈ L ∞ loc (R, H s (R 2 )), s > 5 2 , alors il existe une constante C(t) indépendante de ε ∈]0, 1] tel que

W ε 2 H s ≤ C(t){ W 0 2 H s + S 2 L ∞ ([0,t],H s ) } (0.0.15) Corollaire 0.0.1. Si W 0 ∈ H s (R 2 ) et S ∈ L ∞ loc (R, H s (R 2 )), s > 5 2
, alors le système (0.0.14) admet une solution W ∈ H s (R 2 ) dans l'intervalle de temps [0,T].

Théorème 0.0.3. Si W 0 ∈ H s (R 2 ) et S ∈ L ∞ loc (R, H s (R 2 )), s > 5 2
, alors le système (0.0.14) admet une solution unique dans H s (R 2 ).

Dans la dernière section, nous adoptons une méthode de volumes finis pour résoudre les équations de Saint Venant (0.0.1) combinée à un algorithme de simplexe ou un algorithme de gradient projeté pour la résolution du problème d'optimisation de forme et minimiser la fonction objective (0.0.9). Dans le deuxième chapitre, nous nous intéressons aux équations de Navier-Stokes non stationnaires (0.0.3). Le but de cette étude et d'adopter une approche multicouches qui consiste à subdiviser la hauteur de l'eau en plusieurs couches α. Nous notons la hauteur de l'eau dans chaque couche par h α et par u H,α la vitesse horizontale. Dans la première et la deuxième section, nous donnons une formulation mathématique du problème puis nous dérivons le système multicouche bidimensionnel (0.0.4). La troisième section est concernée par l'étude de l'existence et l'unicité du système multicouches 2-D. Afin d'étudier l'existence et l'unicité de la solution du système (0.0.4), nous le transformons sous forme d'un système de deux équations l'une est hyperbolique et l'autre est parabolique. Nous considérons U = (u 1 , ..., u N ), le système à étudier s'écrit

                                                     ∂ t h + ∂ x h N β=1 l β u β = -G 1/2 ∂ t U -ν∂ xx U = -g∂ x h(1, ..., 1) T -g∂ x z B (1, ..., 1) T - 1 ρ ∂ x p S (1, ..., 1) T +G 1/2 U h + N β=1 l β ∂ x u β U + N β=1 l β u β U h ∂ x h -∂ x (U 2 ) - U 2 h ∂ x h + (TU )∂ x U + (TU )U ∂ x h h -ν U h ∂ xx h + ν 2 (VU ) ∂ xx z B h + ν 2 (WU ) ∂ xx h h - ν 2 (V∂ x U ) ∂ x z B h - ν 2 (W∂ x U ) ∂ x h h + 2ν (XU ) h 2 + 2ν (YU ) h 2 -G 1/2 (ZU ) h (0.0.16)
où T,V,W,X,Y, et Z sont des matrices données et elles dépendent de h, U, z B . Dans cette partie, nous montrons l'existence et l'unicité de la solution du système (0.0.16). Pour cela, nous montrons ce théorème Notation 0.0.1.

1. Soient U ∈ C(0, T, H 1 (R)) et h ∈ C(0, T, H 1 (R)), on définit la norme (U, h) 2 1_1 = U 2 1 + h 2 1 2. Soient U ∈ C(0, T, H 1 (R)) et h ∈ C(0, T, H 2 (R)), on définit la norme (U, h) 2 1_2 = U 2 1 + h 2 2
3. Soient U ∈ C(0, T, H 2 (R)) et h ∈ C(0, T, H 2 (R)), on définit la norme

(U, h) 2 2_2 = U 2 2 + h 2 2
4. Soient U ∈ C(0, T, H 2 (R)) et h ∈ C(0, T, H 3 (R)), on définit la norme

(U, h) 2 2_3 = U 2 2 + h 2 3
Théorème 0.0.4. On considère le système (0.0.16) avec la condition initiale

(U, h)(0, x) = (U 0 (x), h 0 (x)) ∈ H 2 (R) × H 3 (R)
On suppose z b , p S ∈ H 2 (R), et on note

E = 2 (U 0 , h 0 ) 2_3
On suppose qu'il existe une constante positive η 0 telle que inf x∈R h 0 (x) ≥ η 0 > 0. Alors il existe T > 0 telle que le système (0.0.16) admet une solution forte (U,h) vérifiant

1. U ∈ C(0, T, H 2 (R)) ∩ C 1 (0, T, L 2 (R) ∩ L 2 (0, T, H 3 (R)) 2. h ∈ C(0, T, H 2 (R)) ∩ C 1 (0, T, H 1 (R)) 3. ∀t ∈ [0, T ] ∀x ∈ R, h(t, x) ≥ inf x∈R h 0 (x) max (U, h)(t, .) 2_2 , T 0 U (τ, .) 2 3 dτ 1 2

≤ E

La preuve de ce théorème est divisé en trois sous sections; l'estimation des termes sources, l'étude du système linéarisé, et la construction d'une suite convergente vers la solution du système (0.0.16). Nous écrivons le système (0.0.16) sous la forme compacte

∂ t U -ν∂ xx U = S = S b + S h + S p S + S nl ∂ t h + ū∂ x h = F = -G 1/2 -∂ x (ū)h (0.0.17) avec          ū = N β=1
l β u β , S b = -g∂ x z B (1, ..., 1), S h = -g∂ x h(1, ..., 1), S p S = -g∂ x p S (1, ..., 1)

S nl = S (1)
nl + S

nl + S

nl + S

nl + S (5) nl + S [START_REF] Alvarez-Vazquez | An optimal shape problem related to the realistic design of river fishways[END_REF] nl + S [START_REF] Lions | Optimal Control of Systems governed by partial differential equations[END_REF] nl + S [START_REF] Lions | Optimal control of systems governed by partial differential equations[END_REF] nl + S [START_REF] Rajaratnam | Hydraulics of vertical slot fishways[END_REF] nl + S

nl = ν 2 ((W -2I N )U ) ∂ xx h h ; S (7) nl = ν 2 VU ∂ xx z B h ; S (8) nl = - ν 2 VU ∂ x z B h ; S (6)
nl = -

ν 2 W∂ x U ∂ x h h ; S (10)
nl = 2ν(X + Y) U h 2 Dans ce qui suit, nous supposons que le fond est non pénétrable: l'échange de masse au fond est nulle (G 1/2 = 0). Proposition 0.0.1. On suppose U(t,.),h(t,.) ∈ H 2 (R) tel que h ≥ η 0 , avec η 0 est une constante donnée strictement positive. alors on a • Soient S,F définis dans (0.0.17), alors S,F ∈ H 1 (R), et nous avons les estimations suivantes

S 1 ≤ C(1+ (U, h) 2_3 + (U, h) 2 2_3 + (U, h) 3 2_3 ) (U, h) 2_3 + g z B 2 + 1 ρ p S 2
(0.0.18)

F 1 ≤ C (U, h) 2 2_3 (0.0.19) • Soient (U, h), (U ′ , h ′ ) ∈ H 2 (R) × H 3 (R) tels que (U, h) 2_3 ≤ E, (U ′ , h ′ ) 2_3 ≤ E, et h, h ′ ≥ η 0 > 0 (0.0.20)
où E, et η 0 sont des constantes, alors

S(U, h) -S(U ′ , h ′ ) 1 ≤ C(1 + E + E 2 + E 3 ) (U -U ′ , h -h ′ ) 2_3 (0.0.21) F (U, h) -F (U ′ , h ′ ) 1 ≤ C M E (U -U ′ , h -h ′ ) 2_3 (
(R)), et ū ∈ L ∞ (0, T ; H 2 (R)), h 0 ∈ H 2 (R) Alors 1.
Le problème (Hyp) admet une solution forte h telle que:

h ∈ C(0, t; H 1 (R)) ∩ C 1 (0, T ; L 2 (R)) (0.0.26) 2. Pour tout t ∈ [0, T ], on a l'inégalité d'énergie: h(t, .) 1 ≤ e C 1 Eūt h 0 1 + t 0 e -C 1 Eūt F (τ, .) 1 dτ (0.0.27) avec C 1 > 0, est une constante donnée et Eū := sup 0≤t≤T ū 2 Proposition 0.0.3. Soit T > 0 tel que F ∈ C(0, T ; H 2 (R)), ū ∈ L ∞ (0, T ; H 2 (R) et h 0 ∈ H 2 (R) Alors 1.
Le problème (Hyp) admet une solution forte h telle que: Á la fin de la preuve du théorème 0.0.4, nous construisons une suite récursive (U (j) , h (j) ) = (u

h ∈ C(0, t; H 2 (R)) ∩ C 1 (0, T ; H 1 (R)) (0.0.28) 2. Pour tout t ∈ [0, T ] on a l'inégalité d'énergie: h(t, .) 2 ≤ e C 2 Eūt h 0 2 + t 0 e -C
(j) 1 , ..., u (j)
N , h (j) ) j∈N comme suit

• ∀(t, x) ∈ [0, T ] × R, U (0) , h (0) (t, x) = (U 0 , h 0 )(x) ∈ H 2 (R)
• ∀j ∈ N, (U (j+1) , h (j+1) ) est solution de

         ∂ t U (j+1) -ν∂ xx U (j+1) = S(j) dans [0, T ] × R ∂h (j+1) -ū(j) ∂ x h (j+1) = F (j,j+1) dans [0, T ] × R (U (j) , h (j) )(0, .) = (U (0) , h (0) )(.) dans R (0.0.30)
où pour tout j ∈ N:

   S (j) = S(U (j) , h (j) , ∂ x U (j) , ∂ x h (j) , ∂ xx h (j) ), F (j,j+1) = -G 1/2 -∂ x (ū (j+1) )h (j)
nous définissons les constantes

E 0 = 2 (U 0 , h 0 ) 2,3 et η 0 = 1 2 inf x∈R h 0 (x) Proposition 0.0.4. Soit (h 0 , U 0 ) ∈ H 3 (R) × H 2 (R).
Alors il existe T > 0 petit tel que

• La suite (U (j) , h (j) ) j∈N est bien définie et satisfait, pour tout t ∈ [0, T ] et tout j ∈ N:

U (j) ∈ C (0, T, H 2 (R)) ∩ C 1 (0, t; L 2 (R)) ∩ L 2 (0, T ; H 3 (R)) h (j) ∈ C(0, T ; H 2 (R)) ∩ C 1 (0, T ; H 1 (R)) (0.0.31) • ∀(t, x) ∈ [0, T ] × R et ∀j ∈ N, on a (U (j) , h (j) )(t, .) 2_2 ≤ E 0 t 0 U (j) (τ, .) 2 3 dτ 1/2 ≤ E 0 (0.0.32) h (j) (t, x) ≥ η 0 > 0 (0.0.33)
Après l'étude du système multicouches 2-D avec l'existence et l'unicité de sa solution, Nous généralisons l'approche mutlicouches aux équations de Navier Stokes hydrostatiques non stationnaires 3-D dans la quatrième section

∇.u = 0 ∂U ∂t + ∇ x .(U ⊗ U ) + ∂(U w) ∂z + ∇ x p = µ ∂ 2 U ∂z 2 ∂p ∂z = -g (0.0.34)
avec les conditions aux bords

w(t, x, z b ) = 0 µ ∂U ∂z (t, x, z b ) = kU (t, x, y, z b ) ∂U ∂z (t, x, h) = 0 p(t, x, h) = 0 (0.0.35) et la condition cinématique ∂h ∂t + U (t, x, h)∇ x h -w(t, x, h) = 0 (0.0.36) où u(t, x, z) = (u, v, w) est le vecteur vitesse; et U (t, x, z) = (u, v) est la vitesse horizontale.
Nous dérivons un système multicouches à partir du système (0.0.34). Ce système est utilisé pour la modélisation des structures de passe à poissons. Pour cela, nous considérons le système multicouches suivant:

∂h α ∂t + ∇ x .q α = 0 dans Ω α × [0, T ] ∂q α ∂t + ∇ x .( q α h α ⊗ q α ) + g 2 ∇ x (h α h) = g 2 h 2 ∇ x h α h -gh α ∇ x z b -k α q α h α +2ν α h α q α+1 -h α+1 q α h 2 α+1 h α + h 2 α h α+1 -2ν α-1 h α-1 q α -h α q α-1 h 2 α h α-1 + h 2 α-1 h α f or α = 1, ..., N dans Ω α × [0, T ] (0.0.37)
avec les conditions aux bords et initiales suivantes

h α (0) = h 0 , q α (0) = q 0 dans Ω α q α .n = 0, curl q α h α = 0 sur γ 0 × [0, T ] q α = q 1 n sur γ 1 α × [0, T ] h α = h 2 sur γ 2 α × [0, T ] (0.0.38)
Ensuite, nous dérivons un système adjoint à partir du système (0.0.37),

- ∂p ∂t + 1 h 2 α (q α .∇)r.q α + g∇h.r + g∇z b .r -k α q α h 2 α .r -2ν α h 3 α+1 q α -h 2 α h α+1 q α+1 + 2h α h 2 α+1 q α (h 2 α+1 h α + h 2 α h α+1 ) 2
.r

+ 2ν α-1 2h 2 α h α-1 q α-1 -2h α h 2 α-1 q α -h α h 2 α-1 q α-1 -h 3 α-1 q α (h 2 α h α-1 + h 2 α-1 h α ) 2 .r = - q α h α -v q α h 2 α -σcurl(curl( q α h α )). q α h 2 α dans Ω α × [0, T ] - ∂r ∂t -∇p - 1 h α (q α .∇)r - 1 h α (∇r) t q α + k α r h α + 2ν α h α+1 r h 2 α+1 h α + h 2 α h α+1 + 2ν α-1 h α-1 r h 2 α h α-1 + h 2 α-1 h α = 1 h α ( q α h α -v) + σ 1 h α curl(curl( q α h α )) dans Ω α × [0, T ] (0.0.39)
avec les conditions finales et aux bords suivantes

p(T ) = 0, r(T ) = 0 dans Ω α r.n = 0 sur γ 0 × [0, T ] - q 1 h 2 α .r.n = 0 sur γ 1 α × [0, T ] (p + 1 h 2 (q α .r))n + 1 h 2 (q α .n).r - σ h 2 curl( q α h 2 )τ = 0 sur γ 2 α × [0, T ] (0.0.40)
et nous calculons la dérivée du domaine de la fonction objective à chaque couche α:

j α = 1 2 T 0 Ωα q α h α -v 2 dtdx + σ 2 T 0 Ωα | curl( q α h α ) | 2 dtdx (0.0.41)
Pour le calcul de la dérivée du domaine de la fonction coût (0.0.41), nous montrons ce résultat Théorème 0.0.5. Soit V ∈ Lip(Ω α ; R 2 ), alors la dérivée du domaine de la fonction j α par rapport à Ω α ∈ X 0 α est donnée par

∂ ∂Ω α j α (Ω α ).V = ∂ ∂Ω α J α (Ω α ; h α , q α ).V - ∂ ∂Ω α A α (Ω α ; h α , q α ; p, r).V (0.0.42) avec J α (Ω α , h α , q α ) = 1 2 T 0 Ωα q α h α -v 2 J 1 α (Ωα,hα,q α ) + σ 2 T 0 Ωα | curl( q α h α ) | 2 J 2 α (Ωα,hα,q α ) et A α (Ω α , h α , q α ; p, r) = T 0 Ωα ∂h α ∂t p A 1 α (Ωα,hα,q α ;p,r) + T 0 Ωα (∇ x .q α )p A 2 α (Ωα,hα,q α ;p,r) + T 0 Ωα ∂q α ∂t .r A 3 α (Ωα,hα,q α ;p,r) + T 0 Ωα ∇ x .( q α h α ⊗ q α ).r A 4 α (Ωα,hα,q α ;p,r) + T 0 Ωα g 2 ∇ x (h α h).r A 5 α (Ωα,hα,q α ;p,r) - T 0 Ωα g 2 h 2 ∇ x h α h .r
A 6 α (Ωα,hα,q α ;p,r)

+ T 0 Ωα gh α ∇ x z b .r A 7 α (Ωα,hα,q α ;p,r) + T 0 Ωα k α q α h α .r A 8 α (Ωα,hα,q α ;p,r) - T 0 Ωα 2ν α h α q α+1 -h α+1 q α h 2 α+1 h α + h 2 α h α+1 .r A 9 α (Ωα,hα,q α ;p,r) + T 0 Ωα 2ν α-1 h α-1 q α -h α q α-1 h 2 α h α-1 + h 2 α-1 h α .r
A 10 α (Ωα,hα,q α ;p,r) où (h α , q α ) est solution du système d'état (0.0.37) avec les conditions intiales et aux bords (0.0.38), et (p, r) est solution du système adjoint (0.0.39) avec les conditions finales et aux bords (0.0.40).

La dernière sections est dédiée aux méthodes numériques utilisées. Nous comparons deux techniques de résolution pour le problème d'optimisation de forme; la première est une méthode qui ne s'appuie pas sur l'utilisation du gradient de la fonction objective dans ses itérations. la deuxième technique est de type gradient, Elle fait une appel du gradient de la fonction coût à chaque itération. Enfin, nous donnons quelques exemples de tests numériques.

Quant au troisième chapitre, nous nous intéressons aux équations de Brinkman Forchheimer stationnaires (0.0.5). L'étude dans ce chapitre est faite en plusieurs sections. Tout d'abord, nous donnons une définition des espaces à utiliser pour la suite. Pour d=2, 3 nous définissons

X = (H 1 0 (Ω)) d , Y = L 2 (Ω), Ŷ = {q ∈ H 1 (Ω)} , M = q ∈ L 2 (Ω); Ω qdx = 0
et X ′ est l'espace dual de X. Dans la première section, nous montrons l'existence et l'unicité d'une solution du problème (0.0.5), puis nous introduisons le problème perturbé (1.0.6). L'existence d'une solution faible du problème pénalisé est prouvée en utilisant une méthode de Féado-Galerkin. Théorème 0.0.6. Soit Ω un domaine borné dans R n (n ≥ 0), f ∈ X ′ est une fonction donnée. alors le problème perturbé (1.0.6) admet une solution qui satisfait

ν 2 ∇u ε 2 +a u ε α+2 L α+2 +ε p ε 2 ≤ 1 2ν f 2 X ′ (0.0.43)
Á la fin de cette section, nous établissons les estimations d'erreur de la vitesse et de la pression en norme H 1 et L 2 respectivement. Théorème 0.0.7. Soit (u ε , p ε ) solution du problème (1.0.6) et (u,p) est la solution dans X × Y /R du système (0.0.5). Alors (p ε ) ∈ Y est borne unif ormment en ε (0.0.44)

et quand ε -→ 0. u ε → u dans X (0.0.
où kerB * = {q ∈ Y |(q, ∇.v) = 0 f or all v ∈ X}
Dans la deuxième section, nous entamons l'analyse du problème du contrôle optimal. Le problème d'optimisation de forme consiste à minimiser la fonction suivante

J(Ω) = 1 2 Ω |u ε -u d | 2 dx + σ 2 Ω |curl(u ε )| 2 dx (0.0.
-ν∆v + (∇u ε ) T .v -(u ε .∇)v - 1 2 ∇(u ε .v) + 1 2 (∇.u ε )v + a|u ε | α-2 (|u ε | 2 v + α(u ε .v)u ε ) +∇q = (u ε -u d ) -σ curl(curl(u ε )) dans Ω ∇.v + εq = 0 dans Ω v = 0 sur Γ 0 ∪ Γ 1 ν ∂v ∂n + 2(u ε .n)v -nq = σcurl(u ε ).τ sur Γ 2
(0.0.49) La deuxième partie de cette section est concernée au calcul du gradient de forme de la fonction objective (0.0.48) en fonction des variables d'état u ε , p ε et les variables adjointes v, and q

∇J = 1 2 |u ε -u d | 2 + σ 2 |curl(u ε )| 2 + ∂u ε ∂n . 2ν ∂v ∂n -σcurl(u ε ) ∧ n n (0.0.50)
La quatrième partie de cet axe se focalise sur l'étude d'une méthode pénalisée d'éléments finis. Nous discrétisons le problème (0.0.5), et nous associons au système obtenu le problème discret et perturbé suivant Théorème 0.0.9.

Trouver u ε h ∈ X h tel que ν(∇u ε h , ∇v) + ((u ε h .∇)u ε h , v h ) + 1 2 (∇.u ε h , u ε h .v h ) + a(|u ε h | α u ε h , v h ) + 1 ε I((∇.u ε h )(∇.v h )) = (f, v h ) pour tout v h ∈ X h (0.0.
-u ε h 1 ≤ 3 r -1 + a νr C 1 1 + C β h inf v h ∈X h u -v h 1 + C 2 νr inf q h ∈Y h p -q h + C 3 β h ε p h (0.0.52) et p -pε h Z h ≤ ν(3 -2r) + aC 1 β h 3 r -1 + a νr C 2 1 + C 3 β h inf v h ∈X h u -v h 1 + 1 + C 4 β h 3 r -1 + a νr C 1 inf q h ∈Y h p -q h + C ′′ β 2 h ε p h (0.0.53) où l'espace d'élément finis Z h est défini par Z h = qh ∈ Y h |q h -q h ∈ kerB * h (0.0.54) avec kerB * h = q h ∈ Y h | I(q h ∇.v h ) = 0 ∀v h ∈ X h (0.0.
Supposons u 0 ∈ (H 1 0 (Ω)) d (d = 2, 3), et f ∈ L 2 (Ω), alors ||e(t)|| + t 0 ||e(s)|| 2 1 ds 1 2 ≤ C √ ε ∀ t ∈ [0, T 0 ] Supposons u 0 ∈ (H 1 0 (Ω)) d (d = 2, 3), f ∈ L 2 (Ω) et tf t ∈ L 2 (0, T ; H 1 (Ω)), alors √ t||e(t)|| + t||e(t)|| 1 + t 0 s 2 ||q|| 2 ds 1 2 ≤ Cε ∀ t ∈ [0, T 0 ]
Ce résultat obtenu sera utilisé pour déduire une estimation générale pour l'analyse d'erreur du système non linéaire Théorème 0.0.10. Supposons u 0 ∈ X, f ∈ L 2 (Ω) et tf t ∈ L 2 (0, T ; H 1 (Ω)), alors nous avons les estimations d'erreurs suivantes

√ t||u -u ε || + √ νt||u -u ε || 1 + t 0 s 2 ||p -p ε || 2 d 1 2 ≤ Cε ∀ t ∈]0, T 0 ]
La troisième section est consacrée au contrôle optimal; nous introduisons un problème d'optimisation de forme, nous calculons une équations adjoint associée au problème perturbé (0.0.8) et la fonction coût suivante

J(Ω) = 1 2 T 0 Ω |u ε -u d | 2 dxdt + σ 2 T 0 Ω |curl(u ε )| 2 dxdt (0.0.56) Le système adjoint s'écrit                                    -v t -ν∆v + (∇u ε ) T .v -(u ε .∇)v - 1 2 ∇(u ε .v) + 1 2 (∇.u ε )v + a|u ε | α-2 (|u ε | 2 v + α(u ε .v)u ε ) +∇q = (u ε -u d ) -σ curl(curl(u ε )) dans Ω × [0, T ] ∇.v + εq = 0 dans Ω × [0, T ] v = 0 sur Γ 0 ∪ Γ 1 × [0, T ] ν ∂v ∂n + 2(u ε .n)v -nq = σcurl(u ε ).τ sur Γ 2 × [0, T ]
v(T ) = 0 (0.0.57) Ensuite, nous calculons le gradient de forme de la fonction objective (0.0.56) en focntion des variables d'états et adjointes

∇J = 1 2 |u ε -u d | 2 + σ 2 |curl(u ε )| 2 + ∂u ε ∂n . 2ν ∂v ∂n -σcurl(u ε ) ∧ n n (0.0.58)
Pour ce qui suit, nous définissons les formes suivantes

a(u, v) = ν(A 1/2 u, A 1/2 v), a ε (u, v) = ν(A 1/2 ε u, A 1/2 ε v), c α (u, v) = (a|u| α u, v) ∀u, v ∈ X avec A = -P H ∆ = ∆ est l'opérateur de Stokes et P H est la projection orthogonale dans L 2 (Ω) sur H où H = v ∈ L 2 (Ω), : ∇.v = 0, v.n| Γ = 0 , avec Γ = Γ 0 ∪ Γ 1 ∪ Γ 2
Dans la quatrième section, nous adoptons une méthode d'éléments finis pour discrétiser le problème continu en espace et une schéma d'Euler pour une discrétisation en temps. La formulation variationnelle de l'équation (0.0.7) et de l'équations pénalisée (0.0.8) s'écrit respectivement comme suit

T rouver (u, p) ∈ L ∞ (0, T ; Y ) ∩ L 2 (0, T ; X) × L 2 (0, T ; M ) tel que (u t , v) + a(u, v) -d(v, p) + d(u, q) + b(u, u, v) + c α (u, v) = (f, v), ∀ (v, q) ∈ X × M (0.0.59) et T rouver (u ε , p ε ) ∈ L ∞ (0, T ; Y ) ∩ L 2 (0, T ; X) × L 2 (0, T ; M ) tel que (u εt , v) + a(u ε , v) -d(v, p ε ) + d(u ε , q) + ε(p ε , q) + b(u ε , u ε , v) + c α (u ε , v) = (f, v) ∀ (v, q) ∈ X × M (0.0.60) avec les données initiales suivantes; u(0) = u 0 et u ε (0) = u 0 .
Nous définissons la discrétisation en temps de la formulation faible perturbée (0.0.60) avec le schéma d'Euler

(d t u n ε , v) + a(u n ε , v) -d(v, p n ε ) + d(u n ε , q) + ε(p n ε , q) + b(u n ε , u n ε , v) + c α (u n ε , v) = (f (t n ), v) (0.0.61) pour tout (v, q) ∈ X × M , et 1 ≤ n ≤ N , où 0 < ∆t < 1 est le pas de temps, t n = n∆, t N = T , (u 0 ε , p 0 ε ) = (u 0 , 0) et d t u n ε = u n ε -u n-1 ε ∆t pour 1 ≤ n ≤ N , et d t u 0 ε est défini comme suit (d t u 0 ε , v) = a(u 0 , v) + ((u 0 .∇)u 0 , v) + c α (u 0 , v) = (f (0), v), ∀v ∈ X with divv = 0
Nous supposons que le couple de données (u 0 , f ) vérifie

||Au 0 || + sup t∈[0,T ] {||f (t)|| + ||f t (t)||} ≤ C (A0)
Nous obtenons les résultats suivants Théorème 0.0.11. Supposons que (A 0 ) et cε ≤ 1 sont vérifiées, il existe une constante k 0 > 0 tel que si k 0 ∆t ≤ 1, alors

||A 1/2 ε u m ε || 2 + ∆t m n=1 (||d t u n ε || 2 + ||A ε u n ε || 2 + ||p n ε || 2 + a||u n ε || α+2 L α+2 ) ≤ C (0.0.62) ||d t u m ε || 2 + ||A ε u m ε || 2 + ||p m ε || 2 1 + ∆t m n=1 ||A 1/2 ε d t u n ε || 2 ≤ C (0.0.63)
pour tout 1 ≤ m ≤ N , avec ||.|| 1 est la norme de l'espace de Sobolev H 1 (Ω).

Nous supposons pour le couple (X h , M h ) des espace d'éléments finis associés à X and M, que Il existe un opérateur r h : D(A) ∩ X → X h tel que (A1)

(div(u -r h u), q h ) = 0, ∀q ∈ M, q h ∈ M h (0.0.64) ||A 1/2 (u -r h u)|| ≤ ch||Au||, ||p -π h p|| ≤ ch||p|| 1 , ∀p ∈ H 1 (Ω) ∩ M (0.0.65)
De plus, nous supposons l'inégalité inverse

||A 1/2 v h || ≤ ch -1 ||v h ||, ∀v h ∈ X h (0.0.66) et la condition inf-sup ||q|| ≤ c sup v h ∈X h (div(v h ), q h ) ||A 1/2 v h || , ∀q h ∈ M h (0.0.67)
Théorème 0.0.12. Supposons que (A0) et (A1) et εc 0 ≤ 1, ∆tk 0 ≤ 1 sont vérifiées. Alors on a

||u n εh || 2 + ∆t m n=1 ||A 1/2 u n εh || 2 + a∆t n n=1 u n εh α+2 L α+2 ≤ C, (0.0.68) ||A 1/2 u n εh || 2 + ∆t m n=1 ||d t u n εh || 2 ≤ C + Ch -2 ∆t m n=1 ||A 1/2 (u n ε -u n εh )|| 2 ||u n εh || 2 , +Ch -2 ∆t m n=1 ||u n ε -u n εh || 2 ||A 1/2 u n εh || 2 , (0.0.69) pour tout 1 ≤ m ≤ N.
avec (u n εh , p n εh ) est solution du problème approximé suivant

(d t u n εh , v h ) + a(u n εh , v h ) -d(v h , p n εh ) + d(u n εh , q h ) + ε(p n εh , q h ) + b(u n εh , u n εh , v h ) +c α (u n εh , v h ) = (f (t n ), v h ), ∀(v h , q h ) ∈ X h × M h (0.0.70) avec u 0 εh = r h u 0 , p 0 εh = 0. et (X h , M h
) sont les espaces d'éléments finis associés à X et M. Dans la dernière section, nous implémentons une méthode d'élément finis sous Freefem++ pour résoudre le problème perturbé (0.0.8), combiné à une méthode de gradient projeté pour la résolution du problème d'optimisation de forme afin d'en déduire une forme optimale de la structure de type VSF.

Chapter 1

General introduction

A fluid is a deformable material medium in the case of the action of an external force. It consists of free particles which move relative to each other. For example, liquids are not very compressible fluids; they maintain their volume whatever their forms and present a clean surface. Contrariwise, the gases are compressible fluids. A fluid in a liquid or gaseous state is characterised by a density and a viscosity. The density represents the number of molecules per volume unit. The viscosity characterises the velocity of the fluid movement.

There are many types of flow. In particular, we cite the laminar and turbulent flow. The first type represents parallel straight current lines. The second one is a disorganised flow with the presence of turbulences.

Among the applications of these water flow models are the problems associated to the optimization of fishways structure. In fact, some types of fish accomplish their daily or annual migrations and for long or short distances to complete their life cycles. We are interested in diadromous fishes which migrate between sale and fresh water. These migratory fish species are threatened by urbanization, agriculture, or dams.

Dams are built through centuries by men for agriculture, navigation, electricity or economic reasons. They represent impassable barriers for migratory species for their reproduction or feeding areas during the rise of the rivers. To meet this challenge, man thought to buit fishways for permitting fish to pass with comfortable conditions. Fishways (fish ladder, or fish passage) are hydraulic structures placed near from dams or weirs to allow migratory fishes to cross to upstream river. Most of these channels are built with a slope to the ground and they are divided into some pools. The speed in the channel pools should not be very high to deplete the fish or prevent them from continuing their journeys to the breeding areas. Several types of fish passage are know nowadays. The best references are Clay [START_REF] Clay | Design of fishways and other Fish Facilities[END_REF] for the pool and weir type, Katopodis et al. [START_REF] Katopodis | Denil Fishways of varying[END_REF] for the Denil fishways and for the vertical slot type Rajaratnam et al. [START_REF] Rajaratnam | Hydraulics of vertical slot fishways[END_REF].

In this thesis, we are involved in the third type of fishways: vertical slot fishway (VSF).

VSF has an area of low velocity for fish to rest during upstream passage, a good energy dissipation. This makes them suitable to sites where fish passage is needed at different times of the year and for a maximum number of small species. Naturally, the aim is to find optimal forms of VSF in order to facilitate the crossing of migratory species through dams in convenient conditions.

The goal of a shape design problem is to find an optimal shape domain in a set of admissible domains when a cost functional achieves a minimum or maximum on this domain. The research of shape design is part of optimal control field governed by partial differential equations and it has variety of applications in engineering. In particular, the design of bridges and dams. Recently, Many researcher and mathematicians have been interested in this field of research [START_REF] Lions | Optimal control of systems governed by partial differential equations[END_REF][START_REF] Gunzburger | Perspectives in Flow Control and Optimization[END_REF][START_REF] Wang | Aerodynamic optimization design of centrifugal compressor's impeller with Kriging model[END_REF][START_REF] Mohammadi | Applied Shape optimization for fluids[END_REF][START_REF] Yan | Shape inverse problem for the two-dimensional unsteady Stokes flow[END_REF][START_REF] Maury | Elasto-plastic Shape Optimization Using the Level Set Method[END_REF].

Our purpose is to investigate these structures with different flow models. The first one is based on the shallow water equations. The shallow water systems are a hyperbolic systems, introduced at the end of the nineteenth century by A.J.C Barré de Saint-Venant since 1871 [START_REF] Jean | Théorie du mouvement non permanent des eaux, avec application aux crues des rivières et a l'introduction de marées dans leurs lits[END_REF]. These equations are derived from the incompressible Navier-Stokes equations using the Boussinesq approximation, in which the density variations are neglected everywhere except in the gravity term. They are used to model free-surface geophysical fluids in shallow flows. Due to their numerical efficiency, They are now widely used for the simulation of many current phenomena: environmental pollution, natural disasters, and climate change.

The aim here is to assess the possibility of using a two-dimensional shallow water model to compute the flow pattern in vertical slot fish ladder and deduce an optimal structure allows fish to cross the obstacle in a convenient conditions. We begin by giving a mathematical formulation of channel consisting of ten pools with vertical slots for obtaining a mean flow suitable for large variety of migratory species. The 2-D shallow-water equations with source terms may be written as

∂H ∂t + - → ∇. - → Q = 0 in Ω × (0, T ) ∂ - → Q ∂t + - → ∇.( - → Q ⊗ - → Q H ) + gH - → ∇(H -η) = - → f in Ω × (0, T ) (1.0.1)
Where H is the water depth; -→ u = (u 1 , u 2 ) is the velocity vector; u 1 and u 2 are the x and y components of flow velocity, respectively; Q = (u 1 H, u 2 H) is the unit-width discharge;

η is the bottom geometry; g is the gravitational acceleration and -→ f represents all effects of bottom friction and atmospheric pressure. We complete the 2-D Saint-Venant system with the following initial and boundary conditions

H(0) = H 0 , Q(0) = Q 0 in Ω Q. n = 0, curl Q H = 0 on γ 0 × (0, T ) Q = Q 1 n on γ 1 × (0, T ) H = H 2 sur γ 2 × (0, T ) (1.0.2)
where γ 0 is lateral boundary, γ 1 et γ 2 are the inflow and outflow boundary respectively. The optimum shape design involve an objective function linked to fish's swimming aptitudes.

Firstly, we compute its shape derivative in term of state variables and adjoint variables according to the derivation of an adjoint system. Existence and uniqueness results for the adjoint system are stated using the Lax symbolic symmetrizer for hyperbolic systems and pseudo-differential techniques. A finite volume method is set up to solve the state problem (1.0.1) combined with two minimizing algorithms (Free gradient algorithm and Spectral projected gradient) for the resolution of the shape optimization problem.

The second sort of shallow flow based model is a multilayer model. The multilayer model ovoids the expensive Navier-Stokes equations when the flow is still within the shallow regime.

Starting from the non stationary Navier-Stokes equations

∂ t ρ + ∇.(ρ u) = 0 in Ω × [0, T ] ∂ t (ρ u) + ∇.(ρ u ⊗ u) = ∇.Σ T + ρg in Ω × [0, T ] (1.0.3)
with the stress tensor symbol Σ T is given by

Σ T = -pId + µ(∇ u + (∇ u) ′ )
and where g = (0, -g) ′ ∈ R d is the gravity acceleration, and µ is a given viscosity.

Following the methodology initiated by Audusse and al. [START_REF] Audusse | Numerical simulations of 3D free surface flows by a multilayer Saint-Venant model[END_REF].Then the water height is divide into a number of layers where the hypothesis of Saint-Venant can be applied in each layer. By considering a non constant vertical velocities, the vertical profile of the horizontal velocity is described by the following multilayer system

∂ t h α + ∇ x .(h α u H,α ) = G α+ 1 2 -G α-1 2 . ρ∂ t (h α u H,α ) + ρ∇ x .(h α u H,α ⊗ u H,α ) -∇ x . µh α (∇ x u H,α + (∇ x u H,α ) ′ ) = 1 2 ρG α+ 1 2 ( u H,α+1 + u H,α ) -1 2 ρG α-1 2 ( u H,α + u H,α-1 ) -µ( K α+ 1 2 -K α-1 2 ) -ρgh α ∇ x (z B + h) -h α ∇ x p S .
(1.0.4)

where G α+1/2 is the normal mass flux at the interface between layers α -1 and α.

We prove the existence and uniqueness of a strong solution for the 2-D multilayer system. The proof is done in several stages: the analysis of the linearized problem, the obtaining of an a priori estimates and at the end a Picard scheme for the transition to the nonlinear case. In order to deal with the optimum shape design problem, we proceed with the generalization of the multilayer procedure to the 3-D hydrostatic Navier-Stokes equations and extract a 3-D multilayer system. The multilayer Saint-Venant description of flow allows to derive an optimal fishway structures with respect to the shape of the variable domain for some given cost functionals modeling the ease of fish passage over the structure to their breeding, growing or feeding areas. We derive an adjoint system related to the obtained 3-D multilayer model and express the domain derivative of the cost function in terms of the state and adjoint variables. We present two minimizing procedures: the first one is a free gradient algorithm and the second is a gradient type algorithm namely spectral projected gradient.

The third water flow is based on the stationary porous media model. Fluid flow in porous media are usually described by Darcy Brinkman Forchheimer equations [START_REF] Forchheimer | Wasserbewegung durch boden[END_REF]:

-ν∆u + (u.∇)u + a | u | α u + ∇p = f in Ω; ∇.u = 0 in Ω u = 0 on Γ 0 u = g on Γ 1 ν ∂u ∂n -pn = 0 on Γ 2 (1.0.5)
where u and p present respectively the velocity and the pressure. ν is Brinkman coefficient, a defines Forchheimer coefficient. f denotes the exterior forces and α ∈ [1, 2] is a real number chosen appropriately throughout this work. Γ 0 is the lateral boundary, Γ 1 is inflow, top and bottom flow boundaries, and Γ 2 is the outflow boundary. Firstly, we investigated the well posedness aspect of the problem. We show the existence and uniqueness of a weak solution of the boundary problem (1.0.5).

The importance of the treatment of the incompressibility constraint has been recognized since a long time. Among the method adopted to relax the constraint ∇.u = 0, we cite 1. The penalty method. We refer here to J. Shen [START_REF] Shen | On error estimates of the penalty method for unsteady Navier Stokes equations[END_REF], R. Temam [START_REF] Temam | Une méthode d'approximation de la solution des équations de Navier-Stockes[END_REF] ∇

.u ε + εp ε = 0 in Ω × [0, T ]
2. Artificial compressibility, we mention here the works of Chorin [START_REF] Joel | Numerical solution for the Navier Stokes equations[END_REF] and of R. Temam [START_REF] Temam | Sur l'approximation de la solution des équations de Navier Stokes par la méthode des pas fractionnaires (ii)[END_REF] ∇.

u ε + εp ε t = 0 in Ω × [0, T ]
3. The projection method

∇.u ε -ε∆p ε = 0 in Ω × [0, T ], ∂p ε ∂n = 0 in ∂Ω × [0, T ]
introduced by Temam [START_REF] Temam | Une méthode d'approximation de la solution des équations de Navier-Stockes[END_REF] and Chorin [START_REF] Joel | Numerical solution for the Navier Stokes equations[END_REF][START_REF] Joel | On the convergence of discrete approximations to the Navier Stodes equations[END_REF] and has a high computational cost.

4. The pseudocompressibility method

∇.u ε -ε∆p ε t = 0 in Ω × [0, T ], ∂p ε ∂n = 0 in ∂Ω × [0, T ]
In this work, a penalty method is used to relax the constraint of incompressibility for the velocity. The penalized problem related to the system (1.0.5) writes

-ν∆u ε + (u ε .∇)u ε + 1 2 (∇.u ε )u ε + a | u ε | α u ε + ∇p ε = f in Ω ∇.u ε + εp ε = 0 in Ω u ε = 0 on Γ 0 u ε = g on Γ 1 ν ∂u ε ∂n -p ε n = 0 on Γ 2 (1.0.6)
In order to asses the usefulness of the approach, we establish some a priori estimates helpful to build the error analysis. The existence of a weak solution for the penalized problem (1.0.6) is proved using a Faedo-Galerkin method. The design objective is to minimize a comfort function. We express the shape gradient of the cost function in terms of the velocity value as a state variable, the adjoint variables and the unit normal vector to the boundary of the domain. Nowdays, several works are interested in the finite element method employed in almost domain of engineering and numerical analysis [START_REF] Ciarlet | The Finite Element Method for Elliptic Problems[END_REF][START_REF] Schneider | Highly gravity-driven flow of a NAPL in water-saturated porous media using the discontinuous Galerkin finite-element method with a generalised Godunov scheme[END_REF][START_REF] Zienkiewicz | Finite Element Method in Engineering Science[END_REF]. A discrete finite element method is introduced to approximate the solution of the penalized problem. We establish a priori estimate which proves the convergence of the approximate penalized solution to the solution of the Navier stokes Forchheimer equation. The optimization procedure is implemented using the continuous adjoint method and the finite element method.

The last type of flow model is described by the non stationary Navier-Stokes Forchheimer equations

u t -ν∆u + (u.∇)u + a | u | α u + ∇p = f in Ω ∇.u = 0 in Ω u = 0 on Γ 0 u = g on Γ 1 ν ∂u ∂n -pn = 0 on Γ 2 u| t=0 = u 0 (1.0.7)
We begin by showing the existence and uniqueness of solution for the initial problem (1.0.7).

Error estimates for the velocity and the pressure are established via the energy method.

Next we treat the penalized problem along two parts; the first one is the study of the linearized problem and the second is the treatment of the nonlinear problem.

The perturbed problem is written as

u ε t -ν∆u ε + (u ε .∇)u ε + 1 2 (∇.u ε )u ε + a | u ε | α u ε + ∇p ε = f in Ω ∇.u ε + εp ε = 0 in Ω u ε = 0 on Γ 0 u ε = g on Γ 1 ν ∂u ε ∂n -p ε n = 0 on Γ 2 u ε | t=0 = u 0 (1.0.8)
We introduce a time discretization by the use of a backward Euler scheme combined with fully discrete finite element method to approximate the penalized problem and establish an error estimate for the velocity and the pressure which will be used to show the convergence of the approximate solution to the solution of the initial problem.

The shape optimization problem is to find the shape which is optimal in that it minimizes a comfort cost functional. We derive the adjoint system associated to the penalized problem. We compute the gradient in terms of state and adjoint variables. The optimization procedure is implemented using the continuous adjoint method and the finite element method.

Chapter 2

Optimum Hydrodynamic Design using Shallow Water Equations

Introduction

Many species of salmon, shad, giant catfishes, dorado, sturgeons and eel migrate between the sea and the rivers to complete their life cycle. Free migration routes for fish are crucial to their survival. We take interest in diadromous fish species which immigrate between sale and fresh water.

We distinguish some types of diadromous fish: Anadromous fish (as salmon, smelt, American shad, hickory shad, striped bass, lamprey, gulf sturgeon,...) which live and grow in the salt water, and migrate to freshwater rivers and lakes to reproduce. The Anadromous fish eventually return to freshwater to spawn. About half of all diadromous fish in the world are Anadromous. Adult Catadromous (American eel, European eel, inanga, shortfin eel, longfin eel) live in fresh water, then migrate to the sea for breeding. After hatching, they migrate back to freshwater where they stay until growing into adults. Catadromous fish undertake a great migration from freshwater to spawn in the marine, and they die there due to the effort made for migration. About one quarter of all diadromous fish in the world are catadromous. Amphidromous species (bigmouth sleeper, mountain mullet, sirajo goby, river goby, torrentfish, Dolly Varden) migrate between estuaries and coastal rivers and streams (in both directions). Amphidromous fish live in freshwater for breeding and they leave to the marine for feeding and growing.

The presence of dams without fish passes appears to be a major contributing factor in the decline of migratory species. Fishways have been designed to provide safe passage for migratory species inhabiting the river to get pass towards their breeding or feeding areas.

The utility of such systems has been demonstrated around the world. The best general reference here is Clay [START_REF] Clay | Design of fishways and other Fish Facilities[END_REF] for the pool and weir type, Katopodis et al. [START_REF] Katopodis | Denil Fishways of varying[END_REF] for the Denil fishways and for the vertical slot type Rajaratnam et al. [START_REF] Rajaratnam | Hydraulics of vertical slot fishways[END_REF].

Vertical slot ladders are quite common and use a large narrow slot to control water flow and depths in the pools between slots. This allows fish to swim upstream without leaping over an obstacle. This design reasonably handles the seasonal fluctuation in water levels and is not sensitive to impoundment or upstream water surface elevation changes.

The chapter is devoted to the study of vertical slots fishway. The aim of this work is to assess the possibility of using a two-dimensional shallow water model to compute the flow pattern in vertical slot fish ladder and deduce an optimal structure allows fish to cross the obstacle in a convenient conditions.

The reminder of the chapter is organized as follows. The next section is dedicated to the mathematical formulation and the introduction of the objective function related to the optimal shape design. In sections 3 and 4, we use a method of transport to describe a variational domain in the optimization process. Then the adjoint equations are derived by employing the differentiability of an saddle point problem which includes a Lagrange multiplier function. Section 5 is assigned to the well-posedness of the adjoint system using an approach based on pseudo-differential calculus. The last section provides numerical methods (finite volume scheme combined to minimizing algorithms) with some design examples to prove that our approach could be very useful for the practical purpose.

Mathematical model

A vertical-slot fishway, shown in Figure 1, is a rectangular channel Ω ⊂ R 2 with a sloping floor that is divided into 10 pools by baffles. The pools have a double function: they ensure a proper dissipation of the energy of water flowing through the fish pass, and provide resting areas for the fish. It is worth pointing out that the geometric features of each pool are with a width of 0.97m, a length of 1.213m, also two transition pools, one at the beginning and other at the end of the channel with the same width and a length of 1.5m. Inside each pool, two baffles are built. They have a width of 2r = 0.061m and are vertical to the lateral fishway boundary. The channel is constructed with a slope relative to the ground. The shallow water equations are used to simulate a variety of problems related to environment and coastal engineering. These equations can be obtained by integrating the incompressible Navier-Stokes equations in depth and taking into account the kinematic and kinetic boundary conditions. The 2-D shallow-water equations with source terms may be written as

∂H ∂t + - → ∇. - → Q = 0 in Ω × (0, T ) ∂ - → Q ∂t + - → ∇.( - → Q ⊗ - → Q H ) + gH - → ∇(H -η) = - → f in Ω × (0, T ) (2.2.1)
Where H is the water depth; -→ u = (u 1 , u 2 ) is the velocity vector; u 1 and u 2 are the x and y components of flow velocity, respectively; Q = (u 1 H, u 2 H) is the unit-width discharge;

η is the bottom geometry; g is the gravitational acceleration and -→ f represents all effects of bottom friction and atmospheric pressure.

We introduce three parts of the boundary of Ω: the lateral boundary denoted by γ 0 , the inflow boundary denoted by γ 1 , and the outflow boundary denoted by γ 2 . We take for -→ n the unit outer normal vector to boundary. To obtain a well-posed problem, we add to this system an initial and boundary conditions defined by:

H(0) = H 0 , - → Q (0) = - → Q 0 in Ω, - → Q . - → n = 0 curl - → Q H = 0 on γ 0 × (0, T ), - → Q = Q 1 - → n on γ 1 × (0, T ), H = H 2 on γ 2 × (0, T ) (2.2.
2) The geometry of the vertical slot based on the use of guide elements to lead smooth hydraulic flow into the next slot. The positioning of the guide elements was carried out at two different locations, a and b, which configure the shape of the fish ladder Ω (Figure 2.2). The design variables a and b are subject to constraints in order to ensure a positive influence on the flow in the individual pools. These constraints are formulated as

1 4 1.213 ≤ y 1 , y 3 ≤ 3 4 1.213 0 ≤ y 2 , y 4 ≤ 1 4 0.97 (2.2.3)
In order to provide a comfort conditions during the fish passage and permit to a maximum number of fishes to pass to the river upstream, the following constraints are introduced

y 3 -y 1 ≥ d 1 = 0.1 y 2 -y 4 ≥ d 2 = 0.05 (2.2.4)
A shape optimization problem consists in the minimization of a functional J ∈ R, also called cost function depending on the design variables a and b defining the shape within the admissible constants defining the admissible set X. We have a direct calculation loop for the functional: from a parametrization (a, b) we define a domain Ω(a, b) on which we compute the state equation solution W = (H, Q) and the cost function J(Ω(a, b)):

J : x = (a, b) ∈ X -→ Ω(x) -→ W (Ω(x)) -→ J(x, Ω(x), W (Ω(x)))
We consider that the shape of the structure is efficient if the associated energy dissipation leads to a velocity of water close to a target velocity v related to fishes species and minimizing the flow turbulence in the channel. The target velocity is given by

- → v (x 1 , x 2 ) = (c, 0) if x 2 ≤ 1 3 0.97 (0, 0) otherwise (2.2.5)
Thus, we want to minimize the following cost function

J = 1 2 T 0 Ω - → u -- → v 2 dxdt + α 2 T 0 Ω | rot( - → u ) | 2 dxdt (2.2.6)
with α ≥ 0 is the vorticity parameter, and

- → u = - → Q H where (H, - → Q
) is solution of the shallow water system (2.2.1) with the initial and boundary conditions (2.2.2).

Adjoint system

We consider (p, r) two test functions space in L ∞ (0, T ; H 2 (ω)) 2 . The variational formulation of the state system (2.2.1) is given by

F ind (H, Q) ∈ L ∞ (0, T ; H 2 (ω)) 2 such that A(ω, H, Q; p, r) = L(ω; p, r) ∀(p, r) ∈ L ∞ (0, T ; H 2 (ω)) 2 where A(ω, H, Q; p, r) = ω T 0 { ∂H ∂t + ∇. Q}.p + ω T 0 { ∂ Q ∂t + ∇.( Q H ⊗ Q) + gH ∇(H -η)} r L(ω; p, r) = ω T 0 f . r
We reformulate the objective function as J(ω, H, Q) = j(ω), such that

J(ω, H, Q) = 1 2 ω T 0 Q H -v 2 + α 2 ω T 0 | curl( Q H ) | 2
Theorem 2.3.1. let (H, Q) ∈ L 2 (0, T ; R)×H 2 (0, T ; ω) solution of the system (2.2.1), and (p, r) ∈ L 2 (0, T ; R) × H 2 (0, T ; ω) two test functions, then the adjoint system associated to the state system (2.2.1) take the form

       - ∂p ∂t + 1 H 2 ( Q. ∇) r. Q -gH( ∇. r) -g ∇η. r = -( Q H -v) Q H 2 -α curl(curl( Q H )). Q H 2 - ∂ r ∂t -∇p - 1 H ( Q. ∇) r - 1 H ( ∇ r) t Q = 1 H ( Q H -v) + α H curl(curl( Q H )) (2.3.1)
with final and boundary conditions

                       p(T ) = 0 in ω r(T ) = 0 in ω r. n = 0 in γ 0 × (0, T ) (gH - Q 2 1 H 2 ) r. n = 0 in γ 1 × (0, T ) {p + 1 H 2 ( Q. r)} n + 1 H 2 ( Q. n) r - α H 2 curl( Q H 2 ) τ = 0 in γ 2 × (0, T ) (2.3.2)
46CHAPTER 2. OPTIMUM HYDRODYNAMIC DESIGN USING SHALLOW WATER EQUATIONS Proof. We consider the Lagrangian function (for details see [4], [START_REF] Lions | Optimal Control of Systems governed by partial differential equations[END_REF] given by

L ag (ω, H, Q; p, r) = J(ω, H, Q) -A(ω, H, Q; p, r) + L(ω, H, Q; p, r)
By integration by parts of the function A(ω, H, Q; p, r), deriving in H the function A and J and using the adjoint state (∂ H L ag (ω, H, Q; p, r) = 0), we obtain the following equation: in ω × (0, T )

- ∂p ∂t + 1 H 2 ( Q. ∇) r. Q -gH( ∇. r) -g ∇η. r = -( Q H -v) Q H 2 -α curl(curl( Q H )). Q H 2
with the following final and boundary conditions:

         p(T ) = 0 in ω r. n = 0 in γ 0 × (0, T ) (gH - Q 2 1 H 2 ) r. n = 0 in γ 1 × (0, T )
By integration by parts of the function A(ω, H, Q; p, r), deriving in Q the function A and J and using the adjoint state (∂ Q L ag (ω, H, Q; p, r) = 0), we obtain the following equation: in ω × (0, T )

- ∂ r ∂t -∇p - 1 H ( Q. ∇) r - 1 H ( ∇ r) t Q = 1 H ( Q H -v) + α H curl(curl( Q H ))
with the following final and boundary conditions:

           r(T ) = 0 in ω r. n = 0 in γ 0 × (0, T ) {p + 1 H 2 ( Q. r)} n + 1 H 2 ( Q. n) r - α H 2 curl( Q H 2 ) τ = 0 in γ 2 × (0, T )
Finally, the adjoint system is given by:

       - ∂p ∂t + 1 H 2 ( Q. ∇) r. Q -gH( ∇. r) -g ∇η. r = -( Q H -v) Q H 2 -α curl(curl( Q H )). Q H 2 - ∂ r ∂t -∇p - 1 H ( Q. ∇) r - 1 H ( ∇ r) t Q = 1 H ( Q H -v) + α H curl(curl( Q H )) (2.3.
3) with final and boundary conditions:

                       p(T ) = 0 in ω r(T ) = 0 in ω r. n = 0 in γ 0 × (0, T ) (gH - Q 2 1 H 2 ) r. n = 0 in γ 1 × (0, T ) {p + 1 H 2 ( Q. r)} n + 1 H 2 ( Q. n) r - α H 2 curl( Q H 2 ) τ = 0 in γ 2 × (0, T ) (2.3.4)

Shape derivative

In this section we give a description of the domain derivative of the objective function j.

Let ω 0 be a fixed domain, and we suppose that it is a bounded open subset of R 2 with Lipschitz boundary. We denote by Lip(ω 0 ; R 2 ) the Lipschitz applications domain, which is defined by

Lip(ω 0 ; R 2 ) = {φ : ω 0 → R 2 / ∃k ≥ 0 such that φ(x) -φ(y) ≤ k x -y , ∀(x, y) ∈ ω 0 × ω 0 }
We denote by O 0 the subset of bi-Lipschitz homeomorphisms of ω 0 . It is an open subset of Lip(ω 0 ; R 2 ) and it is defined by

O 0 = {ψ : ω 0 → ψ(ω 0 ), bijective / ψ ∈ Lip(ω 0 ; R 2 ), ψ -1 ∈ Lip(ψ(ω 0 ); R 2 )}
The admissible domain denoted by X 0 is a set of domains ω which are bound by bijective Lipschitz functions to the fixed domain ω 0 . It is given as follow

X 0 = {ω = ψ(ω 0 ) / ψ ∈ O 0 } (2.4.1)
Theorem 2.4.1. Let V ∈ Lip(ω; R 2 ), and if the functions A, J and L are differentiable with respect to the domain ω in the admissible set X 0 , then the domain derivative of j at ω ∈ X 0 takes the form

∂ ∂ω j(ω). V = ∂ ∂ω J(ω; H, Q). V - ∂ ∂ω A(ω; H, Q; p, r). V + ∂ ∂ω L(ω; p, r). V (2.4.2)
where Proof. Let ω ∈ X 0 , such that ω = ψ(ω 0 ) with ψ ∈ O 0 . We consider

∂ ∂ω A(ω; H, Q; p, r). V = T 0 ω ∂H ∂t p( ∇. V ) + T 0 ω ( ∇. Q)p( ∇. V ) - T 0 ω ( ∇ V ) t : ∇ Qp + T 0 ω ∂ Q ∂t . r( ∇. V ) + T 0 ω ( ∇. Q) Q H . r( ∇. V ) - T 0 ω ( ∇ V ) t : ∇ Q Q H . r + T 0 ω ( Q. ∇) Q H . r( ∇. V ) - T 0 ω ( ∇ V Q. ∇) Q H . r + T 0 ω gH ∇H. r( ∇. V ) - T 0 ω gH( ∇ V ) t ∇H. r - T 0 ω gH ∇η. r( ∇. V ), f or V ∈ Lip(ω; R 2 ) ∂ ∂ω L(ω; p, r). V = T 0 ω f . r( ∇. V ), f or V ∈ Lip(ω; R 2 ) ∂ ∂ω J(ω; H, Q).. V = 1 2 T 0 ω || Q H -v|| 2 ( ∇. V ) + 1 2 T 0 ω |curl( Q H )| 2 ( ∇. V ) f or V ∈ Lip(ω; R 2 ),
F ∈ Lip(ω; R 2 ) a bi-Lipschitz homeomorphism such that ω = F (ω) ∈ X 0 . Let V ∈ Lip(ω; R 2 ) such that V = F -I.
We define the function d by d : ω ∈ X 0 → d(ω) ∈ R, and we introduce a transported function called

d d : F ∈ Lip(ω; R 2 ) → d(F ) = d(F (ω)) = d(ω) ∈ R.
Thus, the domain derivative of d at a given ω ∈ X 0 writes

∂ ∂ω d(ω). V = ∂ ∂F d(I). V , f or V ∈ Lip(ω; R 2 ) (2.4.3)
The calculation of the domain derivative of j is done by introducing the transported functions of A, L, and J. In order to realize this computation, we define the function A as a sum of a number of functions as follow

A(ω, H, Q; p, r) = 6 k=1 A k (ω, H, Q; p, r) Explicitly A(ω, H, Q; p, r) = T 0 ω ∂H ∂t p A 1 (ω,H, Q;p, r) + T 0 ω ( ∇. Q)p A 2 (ω,H, Q;p, r) + T 0 ω ∂ Q ∂t . r A 3 (ω,H, Q;p, r) + T 0 ω ∇.( Q H ⊗ Q). r A 4 (ω,H, Q;p, r) + T 0 ω gH ∇(H). r A 5 (ω,H, Q;p, r) + T 0 ω -gH ∇η. r A 6 (ω,H, Q;p, r)
With a similar manner we define J as a sum of

J(ω, H, Q) = 2 k=1 J k (ω, H, Q) Since, J(ω, H, Q) = 1 2 T 0 ω Q H -V 2 J 1 (ω,H, Q) + α 2 T 0 ω | curl( Q H ) | 2 J 2 (ω,H, Q) and L(ω; p, r) = T 0 ω f . r
Next, for computing the domain derivative of each function above, we introduce its transported function. In particular for the first function

A 1 (ω, H, Q; p, r) = T 0 ω ∂H ∂t p, 50CHAPTER 2

. OPTIMUM HYDRODYNAMIC DESIGN USING SHALLOW WATER EQUATIONS

we introduce the transported function

A 1 (F , H, Q; p, r) = A 1 (F (ω), H • F -1 , Q • F -1 ; p • F -1 , r • F -1 ) = A 1 (ω, H, Q; p, r) = T 0 ω ∂H ∂t p | det( ∇F ) | As it is known, F →| det( ∇F ) | is a differentiable application. Its derivative writes ∂ ∂F | det( ∇F ) | (I). V = ∇. V , f or V ∈ Lip(ω; R 2 )
and we have for

V ∈ Lip(ω; R 2 ) ∂ ∂ω A 1 (ω, H, Q; p, r). V = ∂ ∂F A 1 (I, H, Q; p, r). V = T 0 ω ∂H ∂t p( ∇. V )
For the second function A 2 (ω, H, Q; p, r), we consider the following transported function

A 2 (F , H, Q; p, r) = A 2 (F (ω), H • F -1 , Q • F -1 ; p • F -1 , r • F -1 ) = A 2 (ω, H, Q; p, r) = T 0 ω ( ∇F -1 • F ) t : ∇( Q)p | det( ∇F ) |
where the inner product "X:Y" is expressed as

X : Y = 2 k,l=1 X kl Y kl .
Taking into account that the application

F → ∇F -1 •F is differentiable. Its derivative takes the form ∂ ∂F ( ∇F -1 • F )(I). V = -∇. V , f or V ∈ Lip(ω; R 2 ) Then, for V ∈ Lip(ω; R 2 ) ∂ ∂ω A 2 (ω, H, Q; p, r). V = ∂ ∂F A 2 (I, H, Q; p, r). V = T 0 ω ( ∇. Q)p( ∇. V ) - T 0 ω ( ∇ V ) t : ( ∇ Q)p
For the third function A 3 (ω, H, Q; p, r) , we consider the following transported function

A 3 (F , H, Q; p, r) = A 3 (F (ω), H • F -1 , Q • F -1 ; p • F -1 , r • F -1 ) = A 3 (ω, H, Q; p, r) = T 0 ω ∂ Q ∂t . r | det( ∇F ) | Consequently, for V ∈ Lip(ω; R 2 ) ∂ ∂ω A 3 (ω, H, Q; p, r). V = ∂ ∂F A 3 (I, H, Q; p, r). V = T 0 ω ∂ Q ∂t . r( ∇. V )
For the function A 4 (ω, H, Q; p, r) , we consider the following transported function

A 4 (F , H, Q; p, r) = A 4 (F (ω), H • F -1 , Q • F -1 ; p • F -1 , r • F -1 ) = A 4 (ω, H, Q; p, r) = T 0 ω ( ∇F -1 • F ) t : ∇( Q H ⊗ Q). r | det( ∇F ) |
With a similar manner used for the function

A 2 (ω, H, Q; p, r) we obtain for V ∈ Lip(ω; R 2 ) ∂ ∂ω A 4 (ω, H, Q; p, r). V = ∂ ∂F A 4 (I, H, Q; p, r). V = T 0 ω ( ∇. Q) Q H . r( ∇ V ) - T 0 ω ( ∇ V ) t : ( ∇ Q) Q H . r + T 0 ω ( Q. ∇) Q H . r( ∇ V ) - T 0 ω ( ∇ V Q. ∇) Q H . r
Finally, with same calculations, we get the domain derivative of the other functions for

V ∈ Lip(ω; R 2 ) ∂ ∂ω A 5 (ω, H, Q; p, r). V = T 0 ω gH ∇H. r( ∇. V ) - T 0 ω gH( ∇ V ) t ∇H. r ∂ ∂ω A 6 (ω, H, Q; p, r). V = - T 0 ω gH ∇η. r( ∇. V )
For J(ω, H, Q; p, r) and L(ω, H, Q; p, r), we follow the same steps computation above. Then we get for

V ∈ Lip(ω; R 2 ) ∂ ∂ω J 1 (ω, H, Q). V = 1 2 T 0 ω | Q H -V | 2 ( ∇. V ) ∂ ∂ω J 2 (ω, H, Q). V = α 2 T 0 ω | curl( Q H ) | 2 ( ∇. V ) ∂ ∂ω L(ω; p, r). V = T 0 ω f . r( ∇. V )
Finally, By injecting all previous results in (2.4.2), we obtain the formula for the domain derivative of the objective function j at ω ∈ X 0 in terms of (H, Q) solution of (2.2.1)-(2.2.2), and (p, r) solution of (2.3.1)-(2.3.2).

Analysis of the adjoint problem

Pseudo-differential tools

In this section, we present some properties on pseudo-differential operators. We restrict ourselves to the main tools which will be used in the following. We consider ω is a set of R 2 . We define S as the space C ∞ of complex-valued functions defined in R 2 , which decrease faster than any polynomial function of | x | as | x |-→ ∞ (their derivative are so).

The spatial Fourier transform p(ξ) of p(x) ∈ S writes

p(ξ) = R 2 p(x)e -ixξ dx
conversely, the inverse formula takes the form

p(x) = 1 (2π) 2 R 2 p(ξ)e -ixξ dξ
Consider a sequence of scalar α = (α j ) j=1,2 , the α derivative is given by p(D) is called the differential operator with symbol p(ξ) = ξ α . The differential operator depends the space variable too in general. More generally, the differential operator of order k can be written as

D α p(x) = 1 (2π) 2
p(x, D) = |α|≤k a k (x)D α
for some smooth coefficients a α (x). The symbol of this differential operator writes

p(x, ξ) = |α|≤k a α ξ α
For instance, for u ∈ S, we have the following formula

p(x, D)u(x) = F -1 ξ→x (p(x, ξ)û(ξ)) = 1 (2π) 2 R 2 p(x, ξ)û(ξ)e ixξ dξ
There are some restrictions about the differential operators. In particular they are associated only to polynomial symbols. More generally, we can treat a general class of symbols by introducing the pseudodifferential operators. For instance, if p(x, ξ) is a functions in the researched admissible class, it verifies

|∂ α ξ p(x, ξ)| ≤ c α Λ(ξ) m-|α| and |∂ α x p(x, ξ)| ≤ c α Λ(ξ) m
We define the Sobolev space H s (R 2 ), s ∈ R as

H s (R 2 ) = {W ∈ S ′ : (1+ | ξ | 2 ) s 2 Ŵ (ξ) ∈ L 2 (R 2 )} . H s is the usual Sobolev norm based on L 2 , W H s = Λ s W L 2 (R) , with Λ s = (1 -∆) s 2
where ∆ is the Laplace operator. Let p(x, ξ) be a

C ∞ function. p(x, ξ) ∈ S m 1,0 , m ∈ R means that | D β x D α ξ p(x, ξ) |≤ C α,β (1+ | ξ |) m-|α| , ∀α, β ∈ R And p(x, ξ) ∈ S m means that: p(x, ξ) ∼ j≤m p j (x, ξ) 54CHAPTER 2. OPTIMUM HYDRODYNAMIC DESIGN USING SHALLOW WATER EQUATIONS where p j (x, ξ) is such that: for | ξ |≥ 1, p j (x, rξ) = r j p j (x, ξ), | ξ |≥ 1, r ≥ 1
Pseudo-differential operator P defined by:

P(x, D)ψ = p(x, ξ) ψ(ξ)e ixξ dξ (2.5.1)
is called of order m if p(x, ξ) ∈ S m . OP (Σ) denotes the set of operators with symbols in Σ.

Main result

In this section, we establish a conservative form of the adjoint system obtained in the previous section. We present a standard method to construct a symmetrizer of the found system. In order to obtain a conservative form of our adjoint system (2.3.1), we set

Q = (Hu 1 , Hu 2 ) t , r = (r 1 , r 2 ) t and v = (v 1 , v 2 ) t .
We consider the following scalar quanti-

ties; F = ( Q H -v). Q H 2 + α curl(curl( Q H )). Q H 2 , G 1 = 1 H ( u 1 H -v 1 ) + α 1 H (∂ yx u 2 -∂ yy u 1 )
and

G 2 = 1 H ( u 2 H -v 2 ) + α 1 H (∂ xx u 2 -∂ xy u 1 ).
After heavy calculus the system (2.3.1) takes its conservative form

∂ W ∂t + M (H, Q) ∂ W ∂x + N (H, Q) ∂ W ∂y = S(H, Q, W ) (2.5.2) Where W = (p, r 1 , r 2 ) t , c = gH M (H, Q) =    0 c 2 -u 2 1 -u 1 u 2 1 2u 1 u 2 0 0 u 1    , S(H, Q, W ) =    F -g ∇η. r -G 1 -G 2    and N (H, Q) =    0 -u 1 u 2 c 2 -u 2 2 0 u 2 0 1 u 1 2u 2    Definition 2.5.1. The system (2.5.2) is hyperbolic if the matrix E = M (H, Q).n 1 + N (H, Q).n 2 is diagonalizable. Proposition 2.5.1. If u 1 , u 2 , c ∈ R, then the system (2.5.2) is hyperbolic. Proof: The matrix E = M (H, Q).n 1 + N (H, Q).n 2 writes E =    0 (c 2 -u 2 1 )n 1 -u 1 u 2 n 2 -u 1 u 2 n 1 + (c 2 -u 2 2 )n 2 n 1 2u 1 n 1 + u 2 n 2 u 2 n 1 n 2 u 1 n 2 u 1 n 1 + 2u 2 n 2   
which is diagonalizable, and has the following three eigenvalues

λ 1 = u 1 n 1 + u 2 n 2 -c, λ 2 = u 1 n 1 + u 2 n 2 , λ 3 = u 1 n 1 + u 2 n 2 + c
Moreover, The eigenvectors of the matrix E writes

X = 1 2c    u 1 n 1 + u 2 n 2 + c 2(u 1 n 2 -u 2 n 1 ) -(u 1 n 1 + u 2 n 2 ) + c -n 1 -2n 2 n 1 -n 2 2n 1 n 2   
and we have

X -1 =    1 u 1 -c.n 1 u 2 -c.n 2 0 -c.n 2 c.n 1 1 u 1 + c.n 1 u 2 + c.n 2   
2.5.3 Existence and Uniqueness using the pseudo-differential calculus

Construction of symmetrizer

The objective of this subsection is to construct a symmetrizer of the system (2.5.2) . In order to achieve this, we introduce a standard definition of the symmetrizer notion.

Definition 2.5.2. Let R(t, x, W , ξ) be a smooth function on R×R 2 ×R 3 ×R 2 * , homogeneous of degree 0 in ξ. R(t, x, W , ξ) is called a symmetrizer of the system (2.5.2) if R(t, x, W , ξ) is a positive definite matrix, and

R(t, x, W , ξ).{M (H, Q)ξ 1 + N (H, Q)ξ 2 } is self-adjoint for each (t, x, W , ξ 1 , ξ 2 ).

56CHAPTER 2. OPTIMUM HYDRODYNAMIC DESIGN USING SHALLOW WATER EQUATIONS

We define the following operator P

P (t, x, W , ξ) = M (H, Q)ξ 1 + N (H, Q)ξ 2 (2.5.3)
We will diagonalize the matrix P (t, x, W , ξ) and we will use the eigenvectors of its adjoint matrix to define the symmetrizer of the system (2.5.2). In general, it becomes difficult in some case to find a symmetrizer which verifies the properties in the definition above. In the following, we present a standard method to construct a symmetrizer for the system (2.5.2).

Proposition 2.5.2. If u 1 , u 2 , ξ 1 , ξ 2 ∈ R, then the matrix P (t, x, W , ξ) is diagonalizable.
Proof. The matrix P (t, x, W , ξ) has three distinct eigenvalues

α 1 = u 1 ξ 1 + u 2 ξ 2 -c, α 2 = u 1 ξ 1 + u 2 ξ 2 , α 3 = u 1 ξ 1 + u 2 ξ 2 + c
The conjugate operator P (t, x, W , ξ) * of P is diagonalizable, and its eigenvectors write

P 1 =    1 u 1 -c.ξ 1 u 2 -c.ξ 2    , P 2 =    0 -c.ξ 2 c.ξ 1    , P 3 =    1 u 1 + c.ξ 1 u 2 + c.ξ 2   
The symmetrizer of the system (2.5.2) writes

R(t, x, W , ξ) = P 1 P * 1 + P 2 P * 2 + P 3 P * 3 (2.5.4) Explicitly R(t, x, W , ξ) = 2        1 u 1 u 2 u 1 u 2 1 + c 2 2 (ξ 2 1 + 1) u 1 u 2 + c 2 2 ξ 1 ξ 2 u 2 u 1 u 2 + c 2 2 ξ 1 ξ 2 u 2 2 + c 2 2 (ξ 2 2 + 1)        (2.5.5)
Which is self-adjoint, and we note that

RP = P * R = 2      α 2 u 1 α 2 + c 2 ξ 1 u 2 α 2 + c 2 ξ 2 u 1 α 2 + c 2 ξ 1 a b u 2 α 2 + c 2 ξ 2 b c      where a = 1 2 {(2u 2 1 + c 2 (n 2 1 + 1))α 2 + 4u 1 c 2 n 1 }, b = 1 2 {(2u 1 u 2 + c 2 ξ 1 ξ 2 )α 2 + c 2 (u 1 ξ 2 + u 2 ξ 1 )},
and c = 1 2 {(2u 2 2 + c 2 (n 2 2 + 1))α 2 + 4u 2 c 2 n 2 }.
Finally, in order to derive an energy estimate we define the operator Q using the symmetrizer R as:

Q = 1 2 (R + R * ) + KΛ -1 (2.5.6)
where K is selected to make Q a positive definite operator in L 2 .

Lemma 2.5.1. Let the operator P be defined by the expression in (2.5.3).

If Q ∈ C 1 (S 0 ) then |< QP v, v >|≤ C( v ) C 1 v 2 L 2
(2.5.7)

Lemma 2.5.2. (Kato Ponce commutator [START_REF] Kato | Commutator estimates and the Euler and Navier Stockes equations[END_REF]) Let the operator P be defined by the expression in (2.5.3), then

[Λ s , P ]v L 2 ≤ C{ ∇P ∞ v H s-1 + P H s v ∞ } 2.5.3.

Existence and uniqueness

The objective of this section is to prove the existence and uniqueness of the adjoint system solution. In order to assume that, we introduce the Friedrich's mollifier {J ε } ε>0 , and we define the regularized system obtained from (2.5.2) by:

∂ t W ε + J ε M (H, Q)∂ x (J ε W ε ) + J ε N (H, Q)∂ y (J ε W ε ) = J ε .S( W ε ) W ε (0, x) = W (0, x) (2.5.8)
We define the following operators

P ε (.) = J ε M (H, Q)∂ x (.) + J ε N (H, Q)∂ y (.) (2.5.9)
and

g ε ( W ε ) = J ε .S( W ε ) (2.5.10)
The definition of the symmetrizer R and the regularized operator Q ε using R ε , will help us to derive an energy estimate by applying Λ s and symmetrizer (2.5.4) to the system (2.5.8)

to estimate < Λ s W ε , Q ε Λ s W ε > d dt < Λ s W ε , Q ε Λ s W ε > = 2 < Λ s ∂ W ε ∂t , Q ε Λ s W ε > + < Λ s W ε , Q ′ ε Λ s W ε > (2.5.11)
The second term of the right hand side can be raised as the following: from definition 2.5.2 the construction of the symmetrizer R ε gives

|< Λ s W ε , R ′ ε Λ s W ε >| ≤ C( W ε C 1 ) W ε 2 H s
Using the description of the regularized operator Q ε in (2.5.6), we get

|< Λ s W ε , Q ′ ε Λ s W ε >| ≤ C( W ε C 1 ) W ε 2 H s
It remains to deal with the first term of (2.5.11)

< Λ s ∂ W ε ∂t , Q ε Λ s W ε > = < Q ε Λ s J ε P ε W ε , Λ s W ε > + < Q ε Λ s g ε , Λ s W ε >
The second term in right hand side can be raised as

| < Q ε Λ s g ε , Λ s W ε > | ≤ C(|| W ε || H s )|| W ε || H s
Next, we will try to raise the first term. It can be decomposed as

< Q ε Λ s J ε P ε W ε , Λ s W ε > = < Q ε P ε Λ s J ε W ε , Λ s J ε W ε > + < Q ε [Λ s , P ε ]J ε W ε , Λ s J ε W ε > + < [Q ε Λ s , J ε ]P ε J ε W ε , Λ s W ε > (2.5.12)
The operator

[Q ε Λ s , J ε ] is bounded in the H s-1 norm. the term < [Q ε Λ s , J ε ]P ε J ε W ε , Λ s W ε > can be raised by < [Q ε Λ s , J ε ]P ε J ε W ε , Λ s W ε > ≤ C( W ε C 1 ) P ε J ε W ε H s-1 W ε H s
Using Moser's inequality (see [5]) for the term

P ε J ε W ε H s-1 yields < [Q ε Λ s , J ε ]P ε J ε W ε , Λ s W ε > ≤ C( W ε H s ) W ε H s (2.5.13)
The operator [Λ s , P ε ] has order s, and we use the results of lemma 2.5.2 to raise

|< Q ε [Λ s , P ε ]J ε W ε , Λ s J ε W ε >| ≤ C Λ s W ε L 2 { ∇P ε ∞ W ε H s-1 + P ε H s W ε ∞ } Since |< Q ε [Λ s , P ε ]J ε W ε , Λ s J ε W ε >| ≤ C W ε H s { ∇P ε ∞ W ε H s-1 + P ε H s W ε ∞ }
From the previous expression and using the Sobolev embedding we deduce:

|< Q ε [Λ s , P ε ]J ε W ε , Λ s J ε W ε >| ≤ C( W ε H s ) W ε H s (2.5.14)
We use the results of Lemma.5.1 for the first term of the equation (5.4.13)

|< Q ε P ε Λ s J ε W ε , Λ s J ε W ε >| ≤ C( W ε H s ) W ε H s (2.5.15)
The three inequalities (5.4.14)-(2.5.15) and the Cauchy inequality yields

d dt < Λ s W ε , Q ε Λ s W ε > ≤ C(t){ W 2 H s + S 2 L ∞ ([0,t],H s ) } (2.5.16)
Using the previous results, we prove the following theorem

Theorem 2.5.1. Let W 0 ∈ H s (R 2 ) and S ∈ L ∞ loc (R, H s (R 2 )), s > 5 2 , then there exists a constant C(t) independent of ε ∈]0, 1] such that W ε 2 H s ≤ C(t){ W 0 2 H s + S 2 L ∞ ([0,t],H s ) } (2.5.17)
Proof. We observe that the product < ., Q ε . > defines an equivalent norm in H s and use Gronwall's inequality to get

W ε 2 H s ≤ C(t){ W 0 2 H s + S 2 L ∞ ([0,t],H s ) } (2.5.18) Corollary 2.5.1. Let W 0 ∈ H s (R 2 ) and S ∈ L ∞ loc (R, H s (R 2 )), s > 5 2 , then the sys- tem (2.5.2) have a solution W ∈ H s (R 2 ) in the time interval [0,T]. Proof. Let ε ∈]0, 1], Using theorem.5.1 in a Banach space, we have the energy inequality in terms of W ε d dt < Λ s W ε , Q ε Λ s W ε > ≤ C([0, T ε ]){ W ε 2 H s + S L ∞ ([0,T ],H s ) } (2.5.19)
The existence of a function S(t) independent of ε ∈]0, 1] and an interval time [0,T] yields from Gronwall's inequality:

W ε H s ≤ S(t) ∀t ∈ [0, T ] then { W ε , ε ∈]0, 1
]} converges to a limit W solution of the system (2.5.2). The sequence

{ W ε , ε ∈]0, 1]} is bounded in C([0, T ], H s (B(0, r))) and { ∂ W ε ∂t , ε ∈]0, 1]} is bounded in C([0, T ], H s-1 (B(0, r))
) by definition of (2.5.8) with B(0,r) is a ball. And we have the compact inclusion

H s (B(0, r)) ⊂ H s-1 (B(0, r)).
Ascoli's theorem with interpolation inequalities arguments gives the existence of a sub sequence W εj such that

W εj -→ W in C([0, T ], C 1 (B(0, r)))
the limit W verifies the system (2.5.2) in the distribution sense.

Theorem 2.5.2. Let W 0 ∈ H s (R 2 ) and S ∈ L ∞ loc (R, H s (R 2 )), s > 5 2
, the system (2.5.2) has an unique solution in H s (R 2 ).

Proof : We suppose that the problem (2.5.2) has two solutions W 1 et W 2 such that:

∂ W 1 ∂t = -[M (H, Q)∂ x ( W 1 ) + N (H, Q)∂ y ( W 1 )] + S(H, Q, W 1 ) ∂ W 2 ∂t = -[M (H, Q)∂ x ( W 2 ) + N (H, Q)∂ y ( W 2 )] + S(H, Q, W 2 )
(2.5.20)

We define W = W 1 -W 2 , then W satisfies the following equation:

∂ W ∂t = -[M (H, Q)∂ x ( W 1 ) + N (H, Q)∂ y ( W 1 )] W -[M (H, Q)∂ x ( W 1 ) + N (H, Q)∂ y ( W 1 )] W 2 +[M (H, Q)∂ x ( W 2 ) + N (H, Q)∂ y ( W 2 )] W 2 + S(H, Q, W 1 ) -S(H, Q, W 2 ) (2.5.21)
Other side, we have

S(H, Q, W 1 ) -S(H, Q, W 2 ) = s(H, Q, W 1 , W 2 )( W 1 -W 2 )
where

s(H, Q, W 1 , W 2 ) = 1 0 g ′ (τ W 1 + (1 -τ ) W 2 )dτ, with g(.) = S(H, Q, .
)

and

[M (H, Q)∂ x ( W 2 -W 1 ) + N (H, Q)∂ y ( W 2 -W 1 )] = ( W 1 -W 2 ).a(H, Q, W 1 , W 2 )
Consequently

∂ W ∂t = [M (H, Q)∂ x ( W 1 ) + N (H, Q)∂ y ( W 1 )] W + W .a( W 1 , W 2 , D) W 2 the operator [M (H, Q)∂ x (.) + N (H, Q)∂ y (.)
] is strictly hyperbolic and using the calculus of the proof of theorem.5.1 we infer

d dt W 2 L 2 ≤ C(t) W 2 L 2 (2.5.22)
Using Gronwall's inequality we deduce W ≡ 0.

Numerical methods

The shallow water equations are a set of nonlinear hyperbolic equations. The nonlinear character combined with the hyperbolic type of the equations can lead to discontinuous solutions in finite time. In order to formulate simple and robust numerical procedures, the two-dimensional shallow water equations (2.2.1) are cast in conservation form with source terms

∂ t U + ∂ x F (U ) + ∂ y G(U ) = S(U ) (2.6.1)
where

U =    H Hu 1 Hu 2    , F (U ) =    Hu 1 Hu 2 1 + 1 2 gH 2 Hu 1 u 2    , G(U ) =    Hu 2 Hu 1 u 2 Hu 2 2 + 1 2 gH 2    and S(U ) =    0 f 1 + gH∂ x η f 2 + gH∂ y η   

Finite volume method

Finite volume schemes for the shallow water systems consist in using an upwinding of the fluxes. The problem domain is first discretized into a set of triangular cells T i forming an unstructured computational mesh. Let ∆t be the constant time step and define t n = n∆t for n = 0, ..., N . At each discrete time t n , we note U n i the approximated solution value.

Denote by E(U ) = (F (U ), G(U )) the physical fluxes. By integrating the equation (2.6.1) on a triangle T i , we obtain

T i U t + T i ∇.E(U ) = T i S(U ) (2.6.2)
We note n i the normal on the edges of triangle T i . Using the divergence formula

T i ∇.E(U ) = ∂T i E(U )
.n i dΓ. The equation (2.6.2) takes the form

T i U t + ∂T i E(U ).n i dΓ = T i S(U ) (2.6.3)
The term

∂T i E(U ).n i dΓ can be calculated as

∂T i E(U ).n i dΓ = 3 j=1 E ij .n ij .dl ij
The equation (2.6.3) becomes

| T i | U t + 3 j=1 E ij .n ij .dl ij =| T i | S (2.6.4)
where n ij is the normal on the edge T i /T j , E ij are the discrete fluxes on the interface T i /T j and dl ij is the length of the interface T i /T j . Thus, we now have an equation for each cell i of the form

U t = - 1 | T i | 3 j=1 E ij .n ij .dl ij + S (2.6.5)
We make a finite difference approximation to the time derivative to obtain the scheme

U n+1 i = U n i - dt | T i | . 3 j=1 E ij .n ij .dl ij + dt.S (2.6.6)
Finding the value of the fluxes at the interface is of primary importance. A variety of approximation techniques have been developed to allow efficient calculation of the solution to the Riemann problem. The Roe solver is used to evaluate the term

3 j=1 E ij .n ij .dl ij .

A gradient free algorithm

The design variables related to the shape Ω depend on the two positions of the slot

y = (a, b) = (y 1 , y 2 , y 3 , y 4 ) (Figure 2.
2). We redefine the objective function (2.2.6) in the following way Φ 1 : R 4 → R where Φ 1 (y) = J(Ω(y)). The finite volume scheme (2.6.6) yields, for each time t n , an approximated velocity -→

u n i = -→ Q n i H n i
which induces an approximate objective function

Φ1 (y) = ∆t 2 N n=1 e∈T i [ e - → u n i -- → v 2 + α e | curl( - → u n i ) | 2 ] ( 2
Φ(y) = Φ1 (y) + β 10 j=1 max{( - → φ 2 (y)) j , 0} (2.6.9)
with β is a penalty parameter.

Due to the essentially geometric nature of the problem, we propose a direct search technique for solving the discretized control problem. The Nelder Mead "simplex" algorithm is one of the most widely used methods for nonlinear optimization. The method attempts to minimize a scalar-valued nonlinear function using only function values, without any derivative information. The method constructs a sequence of simplices as approximations to an optimal point. To describe Nelder-Mead iterations, we begin with an arbitrary simplex of 5 vertices y 1 , y 2 , ..., y 5 . We evaluate and order our function on these vertices

Φ(y 1 ) ≤ Φ(y 2 ) ≤ ... ≤ Φ(y 5
). The vertex associated to the maximal value is replaced with a new point y(ν) = (1 + ν)y *νy 5 , where y * is the centroid of the convex hull {y 1 , ..., y 4 }. The value of ν is chosen from this set of values:

ν δ = -0.5, ν γ = 0.5, ν α = 1, ν β = 2.
The choice of these values is determined according to the following algorithm Calculate and sort Φ(y 1 ), Φ(y 2 ), ..., Φ(y 5 ) While | Φ(y 5 ) -Φ(y 1 ) | is not sufficiently small, calculate y(ν β ) and

Φ β = Φ(y(ν β )) then a) If Φ β ≤ Φ(y 1 ) then calculate Φ α = Φ(y(ν α )). If Φ β ≤ Φ α , replace y 5 with y(ν α ); otherwise replace y 5 with y(ν β ). Go to (f) b) If Φ(y 1 ) ≤ Φ β ≤ Φ(y 4 ) then replace y 5 with y β and go to (f) c) If Φ(y 4 ) ≤ Φ β ≤ Φ(y 5 ), then calculate φ γ = Φ(y(ν γ )). If Φ γ ≤ Φ β replace y 5 with
x(ν γ ) and go to (f). Otherwise go to (e)

d) if Φ(y 5 ) ≤ Φ β then calculate φ δ = Φ(y(ν δ )). If Φ δ ≤ Φ y 5
, replace y 5 with y(ν δ ) and go to (f). Otherwise go to (e) e) For j = 2, ..., 5, set y j = y 1 + 1 2 (y jy 1 )

f) Resort values of Φ at each resulting vertex

To prevent stagnation at non-optimal point, a modification proposed by Kelley [3] is used. This technique consists to replace the current simplex by a smaller one.

Theorem 2.6.1. Let f a strictly convex function. Then if Nelder-Mead method is applied to f, then no shrink steps are performed.

Proof. Shrink steps occur when the out and inside contraction are tried and fail. Firstly, we consider that the out contraction is taken. Out contraction occurs when

Φ(x n ) ≤ Φ β ≤ Φ(x n+1 ). x(ν γ ) = 1 2 x * + 1 2 x(ν β
) and if we use the convexity of Φ we have:

Φ γ = Φ(x(ν γ )) = Φ( 1 2 x * + 1 2 x(ν β )) < 1 2 Φ(x * ) + 1 2 Φ(x(ν β )) ≤ max{Φ(x * ), Φ β } Φ(x * ) = Φ( 1 n n i=1 x i ). Using the convexity of Φ we have Φ(x * ) ≤ Φ(x n ), then max{Φ(x * ), Φ β } = Φ β . Finally Φ γ ≤ Φ β
and the out contraction is taken (the shrink step is not applied).

Secondly, the inside contraction is performed only if Φ(x n+1 ) ≤ Φ β . x(ν δ ) can be written as convex combination x(ν δ ) = 1 2 x * + 1 2 x n+1 . If we use the convexity of Φ we obtain:

Φ δ = Φ(x(ν δ )) = Φ( 1 2 x * + 1 2 x n+1 ) < 1 2 Φ(x * ) + 1 2 Φ(x n+1 ) ≤ max{Φ(x * ), Φ(x n+1 )} max{Φ(x * ), Φ(x n+1 )} = Φ(x n+1
), then the inside contraction is taken.

For the rest, let Φ l k be the value of Φ on the vertex l (l = 1,...,n+1) and iteration k.

Theorem 2.6.2. Let Φ a bounded functiona from below on R n . Then if the Nelder-Med algorithm is applied to Φ then 1. The sequence {Φ 1 k } is convergent.

2. If only a finite number of shrinks occur, then all the n+1 sequences {Φ l k }, l=1,...,n+1, converge and their limits satisfy

Φ 1 * ≤ Φ 2 * ≤ ... ≤ Φ n+1 * (where Φ l * = lim k→+∞ Φ l k ).
3. If only a finite number of non-shrinks occur, then all the simplex vertices converge to a single point.

Proof All the values of the sequence {Φ 1 k } are bounded by below by zero (Φ is bounded from below by zero). {Φ 1 k } is a decreasing sequence, then it is convergent. For l = 1,...,n, Φ l k ≤ Φ l+1 k , then if we pass to limit

Φ l * ≤ Φ l+1 * ∀l ∈ [1, n]

Spectral projected gradient

The second adopted approach for resolving our shape optimization problem is consisting in use of a gradient technique called Spectral projected gradient (SPG). This method use the projection into a closed and convex Ω 0 subset of R 4 of all the points y ∈ R 4 satisfying (2.2.3)-(2.2.4).

We denote

σ 1 = σ 3 = 1 4 1.213, δ 1 = δ 3 = 3 4 1.213, σ 2 = σ 4 = 0, δ 2 = δ 4 = 1 4
0.97. Then the admissible set Ω 0 is defined as

Ω 0 = {y = (y 1 , y 2 , y 3 , y 4 ) ∈ R 4 : σ i ≤ y i ≤ δ i , i = 1, ..., 4, y 3 -y 1 ≥ d 1 , y 2 -y 4 ≥ d 2 } (2.6.10)
The optimization problem can be reformulated as min y∈Ω 0 j(y).

(2.6.11)

Through the iterations of the appointed algorithm above the gradient of the cost function is called and used before calling the projection function as follow: -Other iterations: Let ȳ be the current point and ỹ teh previous point. Calculate

x = ȳỹ and y = ∇j(ȳ) -∇j(ỹ) . Then, if x T y > 0, take η = x T x x T y ; elsewhere, take η as a fixed positive value. The value of j in this algorithm is computed from the expression (2.6.7). And y = P Ω 0 (z) is the projection of z ∈ R 4 onto Ω 0 . It is calculated by minimizing a quadratic function of the distance of z to Ω 0 as follow

min y∈Ω 0 1 2 y -z 2 2 = min y∈Ω 0 1 2 z T z -z T y + 1 2 y T y (2.6.12) It leads to minimize                min (y 1 ,y 2 ,y 3 ,y 4 ) 4 i=1 1 2 z 2 i -z i y i + 1 2 y 2 i subject to σ i ≤ y i ≤ δ i , i = 1, ..., 4
y 3 -y 1 ≥ d 1 , y 2 -y 4 ≥ d 2 ⇔              min (y 1 ,y 3 ) 1 2 (z 2 1 + z 2 3 ) -(z 1 y 1 + z 3 y 3 ) + 1 2 (y 2 1 + y 2 3 ) subject to σ i ≤ y i ≤ δ i , i = 1, 3 y 3 -y 1 ≥ d 1 ,
and

                       min (y 2 ,y 4 ) 1 2 (z 2 2 + z 2 4 ) -(z 2 y 2 + z 4 y 4 ) + 1 2 (y 2 2 + y 2 4 ) subject to σ i ≤ y i ≤ δ i , i = 2, 4 y 2 -y 4 ≥ d 2 ,
In order to resolve the two quadratic problems, we use the Karush-Kuhn-Tucker (KKT) technique, and the way of resolution of the two above problems is analogous. To release that, we begin by the first quadratic problem and we choose σ = σ 1 = σ 3 , and δ = δ 1 = δ 3 , then from the five constraints, only three are kept. Finally the optimization problem is reformulated as

min (x 1 ,x2) c -(l 1 x 1 + l 2 x 2 ) + 1 2 (x 2 1 + x 2 2 )
subject to x 2 ≤ δ,

x 1 ≥ σ, x 2 -x 1 ≥ d 1 ,
where ,σ, δ, d 1 , c, and l i , i=1,2, are real numbers. We consider

A =    0 -1 1 0 -1 1    , b =    -δ σ d 1    ,
Then the linear constraints write

Ax -v = b, v ≥ 0, where v = (v 1 , v 2 v 3 )
T is the slack variables. The cost function is strictly convex and quadratic in R 2 . Then the optimisation problem has an unique solution which satisfies the KKT conditions

v = Ax -b -l + x = A T w, v ≥ 0, w ≥ 0, v T w = 0,
where l = (l 1 , l 2 , l 3 ), and w = (w 1 , w 2 , w 3 ) is the vector of the Lagrange multipliers associated to the three constraints above. The unique optimal solution writes

x = l + A T w which gives    x 1 = l 1 + w 2 -w 3 x 2 = l 2 -w 1 + w 3
For computing the Lagrange multipliers w i , we resolve the following LCP problem

v = (-b + Al) + AA T w v ≥ 0, w ≥ 0 v T w = 0
Finally, the gradient of the cost function ∇j

(ȳ) = ∂j ȳ1 (ȳ), ∂j ȳ2 (ȳ), ∂j ȳ3 (ȳ), ∂j ȳ4 (ȳ) called
along the algorithm iteration above can be approximated for a fixed ȳ ∈ Ω 0 and t > 0 small enough, by

∂j ∂y i (ȳ) ≈ j(ȳ + te i ) -j(ȳ) t , f or i = 1, ..., 4.

Design examples

The characteristics of the flow in vertical slot fishways depended mainly on the specific pool design, such as the geometry of the pool. We conduct three different designs of vertical slot fishways with I or L shaped baffle.

The two different types of vertical slot fishways (VSF): a standard VSF and a multi-slot VSF were tested with the same physical and numerical conditions. All initial and boundary conditions are taken constants.

H 0 = H(x, 0) = 0.5m, Q 0 = Q(x, 0) = (0; 0)m 2 s -1 , Q 1 = -0.065/0.97 m 2 s -1
, and H 2 = 0.5 m. the target velocity is c = 0.8 m.s -1 . The vorticity parameter is α = 0. The penalty parameter is β = 500. In these experiments, we consider that only the bottom friction stress is applied associated to Chezy coefficient of 57.36. For fish passage with comfortable conditions, we put d 1 = 0.1 and d 2 = 0.05 (see comfort constraints (2.2.4)).

2.6.4.1 Vertical Slot Fishway ("I" shaped baffle-rectangular slots)

The fishway model consisted of a 12.13 m long, 0.97 m wide and 1.213 m deep flume, which contains 10 pools. There are two rectangular vertical slots in each pool with the geometric dimensions that are detailed in figure 3.5 (see [START_REF] Alvarez-Vazquez | An optimal shape problem related to the realistic design of river fishways[END_REF]). In the case of the initial random shape (Figure 2.4), we observe that the main flow followed a curved path as it crosses the slot, and directly hit the opposite wall. Two low velocity recirculation regions appeared in the two sides of the jet. In the optimal shape configuration (Figure 5.1), the curved trajectory of velocity is removed, and the velocity is close to the uniform target velocity v for the optimal points a = (0.5721,0.1487) and b = (0.8786,0.0520). We considered the fishway under study, whose scheme is shown in Figure 3.8. We maintain the same hydrodynamic and numerical conditions as the previous test. The time interval for the simulation was T = 300s.

The results show a recirculation regions flowing in opposite directions for the random Figure 2.6: Geometric characteristics of a pool with "I" shaped baffle initial shape (Figure 3.9). In the case of the optimal shape the two large recirculation regions at both sides of the slot are highly reduced (Figure 3.12). the obtained optimal points are a = (0.6170,0.1477) and b = (0.8792, 0.0554).

   1 4 1.213 ≤ a 1 , b 1 , c 1 ≤ 3 4 1.213 0 ≤ a 2 , b 2 , c 2 ≤ 1 2 0.97 (2.6.13)
We replace the constraints (2.2.4) by the following constraints 72CHAPTER 2. OPTIMUM HYDRODYNAMIC DESIGN USING SHALLOW WATER EQUATIONS

         b 1 -a 1 ≥ d 1 = 0.1 a 2 -b 2 ≥ d 2 = 0.05 a 1 -c 1 ≥ d 3 = 1 2 0.0305 c 2 -a 2 ≥ d 4 = 1 2 0.0305 (2.6.14)
Finally, the objective function conserves the same writing. We collect all six ten constraints defined in (2.6.13)-(2.6.14) in a function Φ 2 : R 6 -→ R 16 defined by:

- → Φ 2 (a 1 , a 2 , b 1 , b 2 , c 1 , c 2 ) = ( 1 4 1.213 -a 1 , 1 4 1.213 -b 1 , 1 4 1.213 -c 1 , a 1 - 3 4 1.21, b 1 - 3 4 1.213 c 1 - 3 4 1.213, -a 2 , -b 2 , -c 2 , a 2 - 1 2 0.97, b 2 - 1 2 0.97, c 2 - 1 2 0.97, 0.1 -b 1 + a 1 , 0.05 -a 2 + b 2 , 1 2 0.0305 -a 1 + c 1 , 1 2 0.0305 + a 2 -c 2 )
(2.6.15) We use the free gradient algorithm defined previously (Nelder-Mead) to minimise the penalty function written as: The main flow first passed the slot, then flowed back after blocking by the "L" shaped baffle.

Φ(y) = Φ1 (y) + β m j=1 max{( Φ 2 (y)) j , 0} (2
In the controlled case (Figure 3.16), the circulation areas near slots are strongly reduced.

The results pointed out that design is better in terms of velocity and flow pattern.

Conclusion

The optimal shape design techniques combined with a robust total variation diminishing scheme for solving the state system can be considered as useful tools for practical fishway design purpose.

Residence in an area of large scale turbulence will cause the fish to become confused, lose equilibrium and have a reduced swimming aptitude. Simulations of variant configurations provide a detailed flow structures in vertical slot fish ladder and allow to identify hydraulic issues and propose an appropriate type of construction. Of the three designs studied, design 3 is recommended for practical use, in terms of velocity and flow pattern.

Chapter 3

Shape optimization of fishways in multilayer flow based on optimal control theory

Introduction

Through centuries, dams are built for electricity, for navigation, or for agricultural reasons. These constructions are considered as barriers for some fish species, especially when crossing to the river upstream. For this, we need to ensure building a specific hydraulic structure to facilitate fish immigration between saline and fresh water called fishways.

Fishways are designed to enable migratory species to cross the river going to their feeding or growing areas. Those structures are useful to control the water flow and to assure a safe passage of small species. Different designs of fish ways exist; the pool and weir type [START_REF] Clay | Design of fishways and other Fish Facilities[END_REF], the Denil fishways [START_REF] Katopodis | Denil Fishways of varying[END_REF], and the vertical slot type [START_REF] Rajaratnam | Hydraulics of vertical slot fishways[END_REF].

In this chapter we are interested in the vertical slot fishways, which are used frequently nowadays in a large range of hydraulic and biologic environments, as the structure is built with some slots vertical to the boundary of the channel, allowing a good control of the flow and the elevation of the water in the areas near from slots. This way of structure design allows migratory fish to cross dams more easily.

Considering flows with large friction coefficients or with important wind effects, the horizontal velocity can hardly be approximated by a vertical constant velocity as in the Saint-Venant system. A multilayer approach can be a good alternative to the computational issues associated with the free surface Navier-Stokes system. This procedure consists to divide the water height to a number of layers where the hypothesis of Saint-Venant can be applied in each layer. The method leads to a precise description of the vertical profile of the horizontal velocity. The multilayer Saint-Venant description of flow allows to derive an optimal fishway structures with respect to the shape of the variable domain for some given cost functionals modeling the ease of fish passage over the structure to their breeding, growing or feeding areas.

The chapter is organized as follows. The section 2 is devoted to the deduction of a general multilayer Saint-Venant system from the incompressible Navier-Stokes equations. In section 3 will look more closely to a particular model under the hypothesis of hydrostatic pressure. Section 4 deals with the analysis of the local in time existence and uniqueness of solution for the 2D multilayer system. A 3D version of the multilayer model is used to examine an optimal control problem applied to fishway structures, and derive the adjoint system of the multilayer system obtained from the hydrostatic hypothesis in section 5. Finally in the last section, we compare a free gradient and a gradient type algorithm with some numerical examples with structures already used for fishway modeling [START_REF] Alvarez-Vazquez | An optimal shape problem related to the realistic design of river fishways[END_REF], [START_REF] Chorda | Twodimensional free surface flow numerical model for vertical slot fishways[END_REF]. The experiments prove a best control of the water flow in the fishway channel.

Multilayer approach

Preliminaries

In this section we are interested in the multilayer model which to be an approximation of the Navier-Stokes equations. This procedure consists in a discretization of the fluid domain into a number of layers and establishing a velocity vector and scalar pressure which are independent of the vertical variable. We suppose that the vertical velocity is linear with respect to the vertical variable z. Let T be a given positive constant. We denote by Ω the fluid domain. The non-stationary Navier-Stokes system in a d-dimensional space (d = 2,3) takes the form:

∂ t ρ + ∇.(ρ u) = 0 on [0, T ] × Ω ∂ t (ρ u) + ∇.(ρ u ⊗ u) = ∇.Σ T + ρg on [0, T ] × Ω (3.2.1)
with the stress tensor symbol Σ T is given by

Σ T = -pId + µ(∇ u + (∇ u) ′ )
and where g = (0, -g) ′ ∈ R d is the gravity acceleration, and µ is a given viscosity. For obtaining a multilayer system, we introduce a vertical discretization along the z direction of the water depth into N ∈ N * layers of thickness h α (t, x) with N+1 interfaces

Γ α+1/2 (t) of equation z = z α+1/2 (t,
x) for α = 0, 1, ..., N (see figure 1). We denote by z b = z 1/2 the equation of the bottom interface Γ b (t) and by z s = z N +1/2 the equation of the free surface interfaces Γ s (t). the thickness of each layer α is

h α = z α+1/2 -z α-1/2
where z α+1/2 = z b + α β=1 h β for α = 1, ..., N . Finally the height of fluid is defined as

h = z s -z b = N α=1 h α .
The boundary of the fluid domain is divided into Γ b (t) the bottom boundary, Γ s (t) the free surface boundary and Θ(t) the vertical inflow/outflow boundary. We set

∂Ω(t) = Γ b (t) ∪ Γ s (t) ∪ Θ(t). The domain flow Ω(t) is consisting of a set of subdomain Ω α (t) in each layer. Let Ω(t) = ∪ N α=1 Ω α (t)
where:

Ω α (t) = (x, z); x ∈ I H (t) and z α-1/2 < z < z α+1/2 ∂Ω α (t) = Γ α-1/2 (t) ∪ Γ α+1/2 (t) ∪ Θ α (t), with Θ α (t) = (x, z); x ∈ ∂I H (t) and z α-1/2 < z < z α+1/2 (3.2.2)
where I H (t) is the projection of the fluid domain onto the horizontal plane. The inflow/outflow boundary can be considered as a set of boundary domains in each layer α.

It is given by the expression

Θ(t) = ∪ N α=1 Θ α (t).
We introduce some notations:

(i) the differential operator is defined as

∇ = (∂ x 1 , ..., ∂ x d-1 , ∂ z ), then we set ∇ := (∂t; ∇) = (∂ t , ∂ x 1 , ..., ∂ x d-1 , ∂ z ), and ∇ = (∂ x 1 , ..., ∂ x d-1 ) 78CHAPTER 3

. SHAPE OPTIMIZATION OF FISHWAYS IN MULTILAYER FLOW BASED ON OPTIM

(ii) we denote by η T,α+1/2 the time-space unit normal vector to the interface Γ α (t) and is given by

η T,α+1/2 = (∂ t z α+1/2 , ∇ x z α+1/2 , -1) ′ 1+ | ∂ t z α+1/2 | 2 + | ∇ x z α+1/2 | 2
(iii) we denote by η α+1/2 the space unit normal vector to the interface Γ α (t), and is defined as

η α+1/2 = (∇ x z α+1/2 , -1) ′ 1+ | ∇ x z α+1/2 | 2
(iv) if a and b are two tensors of sizes (n,m) and (n,p) respectively, (a;b) is the tensor concatenation of a and b of size (n,m+p)

The following section gives some results about the jump condition of the mass and momentum conservation law.

Weak solution with discontinuities

Definition 3.2.1. We consider that the velocity vector u, the pressure p and the density ρ are possibly discontinuous across the interfaces Γ α+1/2 (t) for α = 1, ..., N -1. Then ( u, p, ρ) is a weak solution of (2.1) if we have the following proprieties:

(i) In each layer Ω α (t) the triplet ( u, p, ρ) is a standard weak solution of (2.1).

(ii) ( u, p, ρ) verifies the following conditions at Γ α+1/2 (t), for α = 0, ..., N :

(a) For the mass conservation law,

[(ρ; ρ u)] |Γ α+1/2 (t) . η T,α+1/2 = 0 (3.2.3)
where [(a; b)] |Γ α+1/2 (t) denotes the jump of the pair (a;b) across

Γ α+1/2 (t), [(a; b)] |Γ α+ 1 2 (t) = (a; b) |Ω α+1 -(a; b) |Ωα |Γ α+ 1 2 (t)
(b) For the momentum conservation law,

[(ρ u; ρ u ⊗ u -Σ T )] |Γ α+1/2 (t) . η T,α+1/2 = 0 (3.2.4)
In the following we assume that the velocity vector and the pressure are independent of the vertical variable and the vertical velocity is linear with respect to z. And we write:

u |Ωα(t) := u α := ( u H,α , w α ) ′ , p α = p |Ωα(t)
with

∂ z u H,α = 0, ∂ z p α = 0, ∂ z w α = d α (t, x) (3.2.5)
for a smooth function d α (t, x). Where u H,α and w α , respectively represent the horizontal and vertical velocities on layer α.

In order to satisfy the mass and momentum conservation two jump conditions at the interfaces must be verified (i) Mass conservation jump conditions

The vector u H,α verifies

u + H,α-1/2 (t, x) = u - H,α+1/2 (t, x) = u H,α (t, x) (3
G α+1/2 := G - α+1/2 = G + α+1/2 (3.2.7)
where

   G + α+1/2 = ∂ t z α+1/2 + u H,α+1 .∇ x z α+1/2 -w + α+1/2 G - α+1/2 = ∂ t z α+1/2 + u H,α .∇ x z α+1/2 -w - α+1/2
and G + α+1/2 is the normal mass flux at the interface Γ α+1/2 (t)

(ii) Momentum conservation jump conditions

We suppose that the tensor

D( u) = ∇ u + (∇ u) ′ has an approximation D ± α+1/2 at the interface Γ α+ 1 2
. It verifies the following conditions:

µD ± α+1/2 . η α+1/2 = µ Dα+1/2 . η α+1/2 ± 1 2 G α+1/2 1+ | ∇ x z α+1/2 | 2 [ u] |Γ α+1/2 (t) (3.2.8)
where

Dα+1/2 =       D H u + H,α+1/2 + u - H,α+1/2 2   ∇x    w + H,α+1/2 + w - H,α+1/2 2       ′ +Q H,α+1/2 ∇ x w + H,α+1/2 + w - H,α+1/2 2 + Q ′ H,α+1/2 2Q v,α+1/2       (3.2.9) 80CHAPTER 3. SHAPE OPTIMIZATION OF FISHWAYS IN MULTILAYER FLOW BASED ON OPTIM
The unknown Q is a new variable introduced for approximating the second order derivatives in z of u. The following equation must be satisfied

Q -∂ z u = 0, with Q = (Q H , Q v ) (3.2.10)
In the following, we assume that the fluid is incompressible in all the fluid domain (i.e the density is a positive constant)

Vertical velocity

The horizontal velocity is determined and is independent of the vertical variable z. We use it to define the vertical velocity profile. We consider z ∈ ]z α-1/2 , z α+1/2 [, the integration with respect to z of the first equation of system (3.2.1) gives

w α (t, x, z) = w + α-1/2 (t, x) -(z -z α-1/2 )∇ x . u H,α (t, x), f or α = 1, ..., N
Using the conditions (3.2.7) at the interfaces, we obtain

w + α+ 1 2 = ( u H,α+1 -u H,α ).∇ x z α+1/2 + w - α+1/2 (3.2.11)
The algorithm in follow computes the vertical velocity using the horizontal velocities:

1. Let G 1/2 be known and using (3.2.7) at the bottom, The quantity w + 1/2 writes

w + 1 2 = u H,1 .∇ x z B + ∂ t z B -G 1/2 2. Then, for α = 1, ..., N and z ∈ ]z α-1/2 , z α+1/2 [, we have    w α (t, x, z) = w + α-1/2 (t, x) -(z -z α-1/2 )∇ x . u H,α (t, x) w + α+1/2 = ( u H,α+1 -u H,α ).∇ x z α+1/2 + w - α+1/2 (3.2.12)
where

w - α+1/2 = w α|Γ α+1/2 (t) = w + α-1/2 -h α ∇ x . u H,α .
Then the velocity vector u writes u(t, x, z) |Ωα(t) = u α (t, x, z) for α = 1, ..., N with

u α (t, x, z) = u H,α (t, x), w + α-1/2 (t, x) -(z -z α-1/2 )∇ x . u H,α (t, x) ′ , (3.2.13)
where w + α-1/2 (t, x) comes from (3.2.12).

Hydrostatic pressure multilayer model

Now, we consider the hypothesis of hydrostatic pressure to construct our model. That is expressed as:

p α (t, x, z) = p α+1/2 (t, x) + ρg(z α+1/2 -z) (3.
v ∈ H 1 (Ω α (t)) 3                                0 = Ωα(t) (∇. u α )ϕdΩ Ωα(t) ρg.vdΩ = Ωα(t) ρ∂ t u α .vdΩ + Ωα(t) ρ( u α .∇ u α ).vdΩ + Ωα(t) µ(∇ u α + (∇ u α ) ′ ) : ∇vdΩ - Ωα(t)
p α ∇.vdΩ

+ Γ α+1/2 (t) Σ - T,α+1/2 + p α+1/2 Id . η α+1/2 .vdΓ - Γ α-1/2 (t) Σ - T,α-1/2 + p α-1/2 Id . η α-1/2 .vdΓ, (3.3.3)
We consider that THE velocity-pressure pairs verify (3.2.5), this leads to choose the test functions such that ∂ z ϕ = 0 and

v = (v H (t, x), (z -z B )V (t, x)) ′ (3.3.4)
where v H (t, x) and V(t,x) do not depend on z.

• Mass conservation

We consider a scalar test function ϕ = φ(t, x) ∈ L 2 (Ω α (t)) independent of z. by integrating, projecting on the horizontal axis and taking into account (3.2.7), we obtain the mass conservation laws

∂ t h α + ∇ x .(h α u H,α ) = G α+1/2 -G α-1/2 , α = 1, ..., N. (3.3.5)
Where

G N + 1 2
represents the mass exchange at the free surface, and the quantity G 1/2 represents the mass exchange at the bottom.

• Momentum conservation

We choose test functions v ∈ H 1 (Ω α ) verifying (3.3.4). By respecting this structure of test function and projecting on the horizontal axis, then taking into account (3.2.8), we obtain the horizontal momentum conservation laws, for α = 1, ..., N ,

ρ∂ t (h α u H,α ) + ρ∇ x .(h α u H,α ⊗ u H,α ) -∇ x . µh α (∇ x u H,α + (∇ x u H,α ) ′ ) = 1 2 ρG α+ 1 2 ( u H,α+1 + u H,α ) -1 2 ρG α-1 2 ( u H,α + u H,α-1 ) -µ( K α+ 1 2 -K α-1 2 ) -ρgh α ∇ x (z B + h) -h α ∇ x p S (3.3.6)
where the term K α+ 1 2 comes from the expression (3.2.8) and we have

K α+ 1 2 = 1+ | ∇ x z α+ 1 2 | 2 [ Dα+ 1 2 . η α+ 1 2 ] H , with [
.] H is the horizontal components of the vector. Then

K α+ 1 2 = D H u + H,α+ 1 2 + u - H,α+ 1 2 2 .∇ x z α+ 1 2 -∇ x w + H,α+ 1 2 + w - H,α+ 1 2 2 ′ -Q H,α+ 1 2 (3.3
                 ∂ t h α + ∇ x .(h α u H,α ) = G α+ 1 2 -G α-1 2 . ρ∂ t (h α u H,α ) + ρ∇ x .(h α u H,α ⊗ u H,α ) -∇ x . µh α (∇ x u H,α + (∇ x u H,α ) ′ ) = 1 2 ρG α+ 1 2 ( u H,α+1 + u H,α ) -1 2 ρG α-1 2 ( u H,α + u H,α-1 ) -µ( K α+ 1 2 -K α-1 2 ) -ρgh α ∇ x (z B + h) -h α ∇ x p S . (3.3.8) with K α+ 1 2 comes from (3.3.7).
After the obtaining of the horizontal velocity proprieties and the description of the vertical velocity profile in the general case, we are interested on the study of the multilayer system with a linear axis of the horizontal variable x ∈ R. Some proprieties of existence and uniqueness of solution will be revealed in the following part.

Analysis of the 2D multilayer system

Let us now consider the 2D multilayer system which is defined as a particular case of the d-dimensional multilayer model (3.3.8). We study the existence and uniqueness of solution of this system in three parts; the first one is the estimate of the source terms, Next we treat a linearised system, and finally we use a Picard scheme to construct a convergent sequence to the solution of our model.

Mass conservation

Let us denote by h α the water height in each layer and by h the height of water. And let for α = 1, ..., N , h α = l α h with l α a positive constant wich represents the fraction of the water height,

G α+ 1 2 -G 1/2 = α β=1 (∂ t h β + ∂ x (h β u β )) (3.4.2)
We consider G N + 1 2 = 0, for α = N the continuity equation writes

∂ t h + ∂ x h N β=1 l β u β = -G 1/2 (3.4.3)
We will use the obtained results in the previous subsection to extract the momentum equation.

Momentum conservation

We use the equation (3.4.3) and rewriting (3.4.2) with a sample manner to get

G α+ 1 2 = G 1/2 + α β=1 l β (∂ t h + ∂ x (hu β ) = G 1/2 + α β=1 l β ∂ x (hu β ) - N γ=1 ∂ x (l γ hu γ ) -G 1/2 Consequently G α+ 1 2 = (1 -L α )G 1/2 + N γ=1 ξ α,γ ∂ x (hu γ ), α = 1, ..., N (3.4.4)
where L α := l 1 + ... + l α , and for α, γ ∈ {1, ..., N }:

ξ α,γ := α β=1 (δ βγ -l β )l γ =    (1 -L α )l γ if γ ≤ α, -L α l γ otherwise
where δ βγ is the standard Kronecker symbol. Now using (3.4.1) and the expression (3.4.4), the equation (3.3.6) takes the form

l α ∂ t (hu α ) + l α ∂ x (hu 2 α ) -l α ∂ x (2νh∂ x u α ) + l α ∂ x h 1 ρ ps + g h 2 -l α 1 ρ ps∂ x h + N γ=1 1 2 [(u α + u α-1 )ξ α-1,γ -(u α+1 + u α )ξ α,γ ]∂ x (hu γ ) = -l α g h∂z B -ν K α+ 1 2 -K α-1 2 - 1 2 [((u α + u α-1 )(1 -L α-1 ) -((u α+1 + u α )(1 -L α )]G 1/2 , (3.4.5) for α = 1, ..., N , with L 0 = 0 and        K α+ 1 2 = 2∂ x u α+ 1 2 z α+ 1 2 -∂ z w α+ 1 2 -Q H,α+ 1 2 u α+ 1 2 := u + α+ 1 2 + u - α+ 1 2 2
and w α+ 1 2 :=

w + α+ 1 2 + w - α+ 1 2 2
We define q α variable of the horizontal discharges in the layer Ω α (t), for α = 1, ..., N . It is written as q α = hu α . Using (3.4.3) and (3.4.5) the model to be analysed is the following . Therefore, the expression of the quantity K α+ 1 2 -K α-1 2 can be expressed using the variables h α , u α , z B and Q H,α as follow

                                   ∂ t h + ∂ x h N β=1 l β u β = -G 1/2 ∂ t (q α ) + ∂ x q 2 α 2 + g h 2 2 + 1 ρ p S h -2ν(∂ x q α - q α h ∂ x h) - 1 ρ p S ∂ x h + N γ=1 1 2hl α [(q α + q α-1 )ξ α-1,γ -(q α+1 + q α )ξ α,γ ]∂ x (q γ ) = g h∂z B - ν l α K α+ 1 2 -K α-1 2 - 1 2hl α [(q α + q α-1 )(1 -L α-1 ) -((q α+1 + q α )(1 -L α )]G 1/2 , (3.4
K α+ 1 2 -K α-1 2 = 2∂ x u α+ 1 2 z α+ 1 2 -2∂ x u α-1 2 z α-1 2 -∂ x (w α+ 1 2 -w α-1 2 ) -Q H,α+ 1 2 -Q H,α-1 2 .
From the expression of the vertical velocity,

w α+ 1 2 -w α-1 2 = 1 2 (w + α+ 1 2 + w - α+ 1 2 -w + α-1 2 -w - α-1 2 ) = 1 2 2w - α+ 1 2 + (u α+1 -u α )∂ x z α+ 1 2 -2w + α-1 2 + (u α -u α-1 )∂ x z α-1 2 = 1 2 (u α+1 -u α )∂ x z α+ 1 2 + (u α -u α-1 )∂ x z α-1 2 -2h α ∂ x u α Observing that z α+ 1 2 = z B + L α h we infer that K α+ 1 2 -K α-1 2 = ∂ x (u α+1 + u α )∂ x (z B + L α h) -∂ x (u α + u α-1 )∂ x (z B + L α-1 h) + ∂ x (h α ∂ x u α ) - 1 2 ∂ x ((u α+1 -u α )∂ x (z B + L α h)) - 1 2 ∂ x ((u α -u α-1 )∂ x (z B + L α-1 h)) -Q H,α+ 1 2 -Q H,α-1 2
(3.4.7) Finally, the expression (3.4.7) is expressed as

K α+ 1 2 -K α-1 2 = ∂ 2 x (h α u α ) - 1 2 (u α+1 -u α-1 )∂ 2 x z B - 1 2 (L α u α+1 + l α u α -L α-1 u α-1 )∂ 2 x h + 1 2 ∂ x (u α+1 -u α-1 )∂ x z B + 1 2 ∂ x (L α u α+1 + l α u α -L α-1 u α-1 )∂ x h -Q H,α+ 1 2 -Q H,α-1 2 (3.4.8)
where,

Q H,α+ 1 2 = 2 u α+1 -u α h α l α + h α+1 l α+1
The following section gives some results for the existence and uniqueness of the 2-D multilayer system. We begin by casting the system (3.4.6) into a system composed of two equations; the first one is a hyperbolic equations associated to the height of water, and an parabolic equation associated to the water flow.

Existence and uniqueness of solution model

The definition of the 2D multilayer system permits to study more of its characteristics. In particular, the existence and uniqueness of its solution. In order to achieve this, we 86CHAPTER 3. SHAPE OPTIMIZATION OF FISHWAYS IN MULTILAYER FLOW BASED ON OPTIM consider the following system:

                                         ∂ t h + ∂ x h N β=1 l β u β = -G 1/2 ∂ t (h α u α ) + ∂ x (h α u 2 α ) -∂ x (2νh α ∂ x u α ) + N γ=1 1 2 [(u α + u α-1 )ξ α-1,γ -(u α + u α+1 )ξ α,γ ] ∂ x (hu γ ) = -gh α ∂ x h -gh α ∂ x z B - h α ρ ∂ x p S -ν( K α+ 1 2 -K α-1 2 ) - 1 2 [(u α + u α-1 )(1 -L α-1 ) -(u α+1 + u α )(1 -L α )] G 1/2
(3.4.9)

In order to cast the system as parabolic type system, firstly we notice that

K α+ 1 2 -K α-1 2 = h α ∂ xx u α + 2∂ x h α ∂ x u α + u α ∂ xx h α - 1 2 (u α+1 -u α-1 )∂ 2 x z B - 1 2 (L α u α+1 + l α u α -L α-1 u α-1 )∂ 2 x h + 1 2 ∂ x (u α+1 -u α-1 )∂ x z B + 1 2 ∂ x (u α+1 -u α-1 )∂ x z B + 1 2 ∂ x (L α u α+1 + l α u α -L α-1 u α-1 )∂ x h -Q H,α+ 1 2 -Q H,α-1 2 -Q H,α+ 1 2 -Q H,α-1 2
(3.4.10) Let U = (u 1 , u 2 , ..., u N ), the unknown couple (h,U) satisfy the following system

                                                     ∂ t h + ∂ x h N β=1 l β u β = -G 1/2 ∂ t U -ν∂ xx U = -g∂ x h(1, ..., 1) T -g∂ x z B (1, ..., 1) T - 1 ρ ∂ x p S (1, ..., 1) T +G 1/2 U h + N β=1 l β ∂ x u β U + N β=1 l β u β U h ∂ x h -∂ x (U 2 ) - U 2 h ∂ x h + (TU )∂ x U + (TU )U ∂ x h h -ν U h ∂ xx h + ν 2 (VU ) ∂ xx z B h + ν 2 (WU ) ∂ xx h h - ν 2 (V∂ x U ) ∂ x z B h - ν 2 (W∂ x U ) ∂ x h h + 2ν (XU ) h 2 + 2ν (YU ) h 2 -G 1/2 (ZU ) h (3.4.11)
Where T,V,W,X,Y,and Z are matrices defined as:

T =(T α,β ) with                          T α,β = 1 2l α (ξ α,β -ξ α-1,β ), if α = β T α,β = - 1 2l α ξ α-1,β , if α = β + 1 T α,β = 1 2l α ξ α,β , if α = β -1 T α,β = 0, otherwise, , V =(V α,β ) with                  V α,β = 1 l α , if α = β + 1 V α,β = - 1 l α , if α = β -1 V α,β = 0, otherwise, W =(W α,β ) with                          W α,β = 1, if α = β W α,β = - L α-1 l α , if α = β + 1 W α,β = L α l α , if α = β -1 W α,β = 0, otherwise, X =(X α,β ) with                  X α,β = 1 l α (l α + l α+1 ) , if α = β f or α = 1, ..., N -1 X α,β = - 1 l α (l α + l α+1 ) , if α = β -1 X α,β = 0, otherwise, Y =(Y α,β ) with                  Y α,β = 1 l α (l α + l α-1 ) , if α = β f or α = 2, ..., N Y α,β = - 1 l α (l α + l α-1 ) , if α = β + 1 Y α,β = 0, otherwise, and
88CHAPTER 3. SHAPE OPTIMIZATION OF FISHWAYS IN MULTILAYER FLOW BASED ON OPTIM Z =(Z α,β ) with                    Z α,β = l α , if α = β f or α = 2, ..., N Z α,β = -(1 -L α ), if α = β -1 Z α,β = 1 -L α-1 , if α = β + 1 Z α,β = 0, otherwise, Definition 3.4.1. 1. Let U ∈ C(0, T, H 1 (R)) and h ∈ C(0, T, H 1 (R)) we define the norm (U, h) 2 1_1 = U 2 1 + h 2 1 2. Let U ∈ C(0, T, H 1 (R)) and h ∈ C(0, T, H 2 (R)) we define the norm (U, h) 2 1_2 = U 2 1 + h 2 2 3. Let U ∈ C(0, T, H 2 (R)) and h ∈ C(0, T, H 2 (R)) we define the norm (U, h) 2 2_2 = U 2 2 + h 2 2 4. Let U ∈ C(0, T, H 2 (R)) and h ∈ C(0, T, H 3 (R)) we define the norm (U, h) 2 2_3 = U 2 2 + h 2 3
Theorem 3.4.1. We consider the system (3.4.11) with initial condition

(U, h)(0, x) = (U 0 (x), h 0 (x)) ∈ H 2 (R) × H 3 (R)
We suppose z b , p S ∈ H 2 (R), and we denote

E = 2 (U 0 , h 0 ) 2_3
We suppose that there exists a positive constant η 0 such that inf x∈R h 0 (x) ≥ η 0 > 0. Then there exists T > 0 such that the system (3.4.11) admits an unique strong solution (U,h) satisfying

1. U ∈ C(0, T, H 2 (R)) ∩ C 1 (0, T, L 2 (R) ∩ L 2 (0, T, H 3 (R)) 2. h ∈ C(0, T, H 2 (R)) ∩ C 1 (0, T, H 1 (R)) 3. ∀t ∈ [0, T ] ∀x ∈ R, h(t, x) ≥ inf x∈R h 0 (x) max (U, h)(t, .) 2_2 , T 0 U (τ, .) 2 3 dτ 1 2

≤ E

The proof of this theorem will be detailed in the following subsections. We begin by the estimation of source terms. Next, we treat a linearised multilayer system, and finally, we construct a convergent subsequence which converges to the system solution.

Estimates on the source terms

The system (3.4.11)

cast in the form    ∂ t U -ν∂ xx U = S ∂ t h + ū∂ x h = F (3.4.12)
with

S = S b + S h + S p S + S nl F = -G 1/2 -∂ x (ū)h where                                ū = N β=1 l β u β S b = -g∂ x z B (1, ..., 1
)

S h = -g∂ x h(1, ..., 1)
S p S = -g∂ x p S (1, ..., 1)

S nl = S (1)
nl + S

nl + S

nl + S

nl + S

nl + S

nl + S

nl + S

nl + S

nl + S

(1)

nl = G 1/2 (I N -Z) U h ; S (2)
nl = N β=1 l β ∂ x u β U ; S (3)
nl = N β=1 l β u β ∂ x h h U ; S (4) nl = ((T -2I N )U ) ∂ x U ; S (5) nl = ((T -I N )U ) U ∂ x h h ; S (6) nl = ν 2 ((W -2I N )U ) ∂ xx h h ; S (7) nl = ν 2 VU ∂ xx z B h ; S (8) nl = - ν 2 VU ∂ x z B h ; S (9) nl = - ν 2 W∂ x U ∂ x h h ; S (10) nl = 2ν(X + Y) U h 2 Lemma 3.4.1. (Moser inequality [5]). Let f, g ∈ H s (R) for s ≥ 1.
Then fg ∈ H s (R) and there exists C M > 0 independent of f and g such that:

f g s ≤ C M f s g s (3.4.13)
In what follows, we suppose that the bottom is non penetrable. Then the the mass exchange at the bottom is G 1/2 = 0. We also give some results about the estimates for the source terms Proposition 3.4.1. We suppose that U(t,.),h(t,.) ∈ H 2 (R) such that h ≥ η 0 , with η 0 a given constant. Then we have

• Let S,F defined as in (3.4.12), then S,F ∈ H 1 (R), and we obtain the following estimates

S 1 ≤ C(1+ (U, h) 2_3 + (U, h) 2 2_3 + (U, h) 3 2_3 ) (U, h) 2_3 + g z B 2 + 1 ρ p S 2
(3.4.14)

F 1 ≤ C (U, h) 2 2_3 (3.4.15) • Let (U, h), (U ′ , h ′ ) ∈ H 2 (R) × H 3 (R) such that (U, h) 2_3 ≤ E, (U ′ , h ′ ) 2_3 ≤ E, and h, h ′ ≥ η 0 > 0 (3.4.16)
Where E, and η 0 are constants, then

S(U, h) -S(U ′ , h ′ ) 1 ≤ C(1 + E + E 2 + E 3 ) (U -U ′ , h -h ′ ) 2_3 (3.4.17) F (U, h) -F (U ′ , h ′ ) 1 ≤ C M E (U -U ′ , h -h ′ ) 2_3 (3.4.18)
Proof. Firstly, for more simplicity we denote . H s (R) by . s , f or s ≥ 1 . And . L 2 (R) by .

Estimates on F:

F 1 = -∂ x (ū)h 1 ≤ C ∂ x (ū)h 1 ≤ C (U, h) 2 2 using (3.4.13)
For estimating the variation of F(.,.), we write

∂ x (ū)h -∂ x ( ū′ )h ′ = ∂ x (ū -ū′ )h + ∂ x ( ū′ )h -∂ x ( ū′ )h ′ = ∂ x (ū -ū′ )h + ∂ x ( ū′ )(h -h ′ )
which gives

F (U, h) -F (U ′ , h ′ ) 1 ≤ C M U -U ′ 2 +C M h -h ′
Estimates on S: we begin by estimating S h , S b , and S p S :

S b 1 ≤ g z b 2 S h 1 ≤ g h 2 S p S 1 ≤ 1 ρ p S 2
For S nl , we estimate all its components

U h 2 1 = U h 2 + ∂ x U h 2 and ∂ x U h = h.∂ x U -∂ x h.U h 2 = ∂ x U h - ∂ x h h 2 U Then ∂ x U h 2 ≤ 2 ∂ x U 2 η 2 0 + 1 η 4 0 U 2 ∞ ∂ x h 2
The Sobolev embedding of L ∞ ֒→ H 1 , yields to

∂ x U h 2 ≤ 2 ∂ x U 2 η 2 0 + 1 η 4 0 U 2 1 ∂ x h 2 Since U h 1 ≤ U 1 (1+ h 1 )
With a same approach, we obtain

S (2) nl 1 ≤ C U 2 2 S (3) nl 1 ≤ C(η 0 ) U 2 1 h 2 (1+ h 2 ) S (4) nl 1 ≤ C(T) U 2 2 S (5) nl 1 ≤ C(T) U 2 1 h 2 (1+ h 2 ) S (6) nl 1 ≤ C(ν, W) U 1 ∂ x h 2 (1+ h 1 ) S (7) nl 1 ≤ C(ν, V, η 0 , z B ) U 1 (1+ h 1 ) S (8) nl 1 ≤ C(ν, V, η 0 , z B ) U 1 (1+ h 1 ) S (9) nl 1 ≤ C(η 0 , W, ν) U 2 h 2 (1+ h 2 ) S (10) nl 1 ≤ C(η 0 , ν, X, Y) U 1 (1+ h 1 )
Now let S = S(U, h) and S ′ = S(U ′ , h ′ ), We compute the variation of S(.,.)

S i h -S ′ i h = -g∂ x (h -h ′ ) ∀i = 1, ..., N S (2) nl -S ′ (2) nl = N β=1 l β ∂ x u β (U -U ′ ) + N β=1 l β (∂ x u β -∂ x u ′ β ) U ′ S (3) nl -S ′ (3) nl = N β=1 l β (u β -u ′ β ) U ∂ x h h + N β=1 l β u ′ β ∂ x h h (U -U ′ ) + N β=1 l β u ′ β U ′ h ∂ x (h -h ′ ) + N β=1 l β u ′ β U ′ ∂ x h ′ hh ′ (h ′ -h) S (4) nl -S ′ (4) nl = ((T -2I N )U ) ∂ x (U -U ′ ) + (T -2I N )(U -U ′ ) ∂ x U ′ S (5)
nl -S

′ (5) nl = ((T -I N )U ) U h ∂ x h + (T -I N )U ′ U -U ′ h ∂ x h + (T -I N )U ′ U ′ h ∂ x (h -h ′ ) + (T -I N )U ′ U ′ (h ′ -h) hh ′ ∂ x h ′ S (6) nl -S ′ (6) nl = ν 2 (W -2I N )(U -U ′ ) ∂ xx h h + (W -2I N )U ′ 1 h ∂ xx (h -h ′ ) + (W -2I N )U ′ ∂ xx h ′ h ′ -h hh ′ S (7) nl -S ′ (7) nl = ν 2 V(U -U ′ ) 1 h + (VU ′ ) h ′ -h hh ′ ∂ xx z B S (8) nl -S ′ (8) nl = - ν 2 V(U -U ′ ) 1 h + VU ′ h ′ -h hh ′ ∂ x z B S (9)
nl -S

′ (9) nl = - ν 2 W∂ x (U -U ′ ) ∂ x h h + W(∂ x U ′ ) 1 h ∂ x (h -h ′ ) + W(∂ x U ′ ) ∂ x h ′ h ′ -h hh ′ S (10) nl -S ′ (10) nl = 2ν (X + Y)(U -U ′ ) 1 h 2 + (X + Y)U ′ (h ′ -h)(h ′ + h) h 2 h ′ 2
Then using Lemma 3.4.1, and (3.4.16) yields

S h -S ′ h 1 ≤ C h -h ′ 2 S (2) nl -S ′ (2) nl 1 ≤ E U -U ′ 2 S (3) nl -S ′ (3) nl 1 ≤ C η 0 E 2 U -U ′ 1 + C η 0 E 2 h -h ′ 2 + C η 2 0 E 3 h -h ′ 1 S (4) nl -S ′ (4) nl 1 ≤ 2CE U -U ′ 2 S (5) nl -S ′ (5) nl 1 ≤ 2 C η 0 E 2 U -U ′ 1 + C η 0 E 2 h -h ′ 2 + C η 2 0 E 3 h -h ′ 1 S (6) nl -S ′ (6) nl 1 ≤ C η 0 E U -U ′ 1 + C η 0 E h -h ′ 3 + C η 2 0 E 2 h -h ′ 1 S (7) nl -S ′ (7) nl 1 ≤ C η 0 U -U ′ 1 + C η 2 0 E h -h ′ 1 S (8) nl -S ′ (8) nl 1 ≤ C η 0 U -U ′ 1 + C η 2 0 E h -h ′ 1 S (9) nl -S ′ (9) nl 1 ≤ C η 0 E U -U ′ 2 + C η 0 E h -h ′ 2 + C η 2 0 E 2 h -h ′ 1 S (10)
nl -S ′ (10) nl

1 ≤ C η 2 0 U -U ′ 1 + C η 4 0 E 2 h -h ′ 1
Adding these inequalities, we get (3.4.17).

Study of the linearised problem

Now, We consider the following linearised problem of the system (3.4.11):

   ∂ t U -ν∂ xx U = S( Ũ , h, ∂ x Ũ , ∂ x h, ∂ xx h) := S ∂ t h + ū∂ x h = F (U, h) := F := -G 1/2 -∂ x (ū) h (3.4.19)
where Ũ , and h are known. We separate the problem (3.4.19) into two Cauchy problems

   ∂ t U -ν∂ xx U := S U (0, .) = U 0 (.) ∈ H 2 (R) (Par) and 
  ∂ t h+ ū∂ x h = F h(0, .) = h 0 ∈ H 2 (R) (Hyp)
Proposition 3.4.2. Let S ∈ C(0, T, H 1 (R)) for T > 0.

1. The initial value problem (Par) has an unique strong solution U which satisfies:

U ∈ C(0, T ; H 1 (R)) ∩ C 1 (0, T ; L 2 (R)) ∩ L 2 (0, T ; H 3 (R)) (3.4.

20)

2. There exist two positive constants C 1 and C 2 , depending only on the viscosity such that for any t ∈ [0,T]:

U (t, .) 2 2 +C 1 t 0 U (τ, .) 2 3 dτ ≤ e t U 0 2 2 +C 2 t 0 S(τ, .) 2 1 dτ (3.4.21)
Proof. Existence: We note K t the Green Kernel of the operator ∂ t -ν∂ xx . Then, the solution of problem (Par) is given by

∀(t, x) ∈ [0, T ] × R, U (t, x) = [K t * U 0 ](x) + t 0 K t-s * S(s, .)ds Then U ∈ C(0, T ; H 2 (R))
. Moreover, we note that

94CHAPTER 3. SHAPE OPTIMIZATION OF FISHWAYS IN MULTILAYER FLOW BASED ON OPTIM ∂ t U = ν∂ xx U + S ∈ C(0, T ; L 2 (R))
Consequently

U ∈ C(0, T ; H 2 (R)) ∩ C 1 (0, T ; L 2 (R))
Uniqueness. We multiply the first equation of system (Par) by U and integrate with respect to x, we obtain, for any λ > 0:

1 2 d dt U 2 +ν ∂ x U 2 ≤ λ 2 U 2 + 1 2λ S 2
Derive (Par) with respect to x, multiplying by ∂ x U and integrating in space, yields, for any λ > 0:

1 2

d dt ∂ x U 2 +ν ∂ xx U 2 ≤ λ 2 ∂ x U 2 + 1 2λ ∂ x S 2 .
Finally, we derive (Par) two times with respect to x, multiply by ∂ xx U and integrate in space. We obtain an approximation in terms of derivative of S. After the integration of the source term, we have for any λ > 0,

| R ∂ xx S∂ xx Udx | = | R ∂ x S∂ xxx Udx | ≤ λ 2 ∂ xxx U 2 + 1 2λ ∂ x S 2
Let C 1 := 2νλ > 0. By summing the previous inequalities we obtain:

d dt U 2 2 +C 1 U 2 3 ≤ λ U 2 2 + 1 λ S 2
1 . The Gronwall Lemma gives the estimates (3.4.21). The uniqueness of solution is given by the priori estimate (3.4.21). Lemma 3.4.2. We suppose u ∈ L ∞ (0, T ; H 2 (R)) for T > 0. Then there exits C > 0 such that

∂ xx (u∂ x h) -u∂ xxx h ≤ C sup 0≤τ ≤T u(τ, .) 2 h 2 , ∀h ∈ L ∞ (0, T, H 2 (R))
Proof. We compute the difference

∂ xx (u∂ x h) -u∂ xxx h = 2∂ x (u)∂ xx h + ∂ xx (u)∂ x h
Using the embedding of L ∞ ֒→ H 1 we obtain

(∂ xx (u∂ x h) -u∂ xxx h)(t, .) ≤ C sup 0≤τ ≤T u(τ, .) 2 h(t, .) 2 Lemma 3.4.3. Let T > 0 such that F ∈ C(0, T ; H 1 (R)), and ū ∈ L ∞ (0, T ; H 2 (R)), h 0 ∈ H 2 (R) Then 1.
The problem (Hyp) has an unique strong solution h such that:

h ∈ C(0, t; H 1 (R)) ∩ C 1 (0, T ; L 2 (R)) (3.4.22)
2. For all t ∈ [0, T ] we have the energy inequality for k = 1,2:

h(t, .) 1 ≤ e C 1 Eūt h 0 1 + t 0 e -C 1 Eūt F (τ, .) 1 dτ (3.4.23)
with

C 1 > 0, is a given constant and Eū := sup 0≤t≤T ū 2
Proof. Uniqueness. We multiply (Hyp) by h and integrate with respect to x we obtain

1 2 d dt h 2 = - R ū∂ x ( h 2 2 ) + R F h
Using the Holder inequality applied to the right hand side and integrating by parts its first term, we get

1 2 d dt h 2 ≤ 1 2 Eū h 2 + F h
We divide the previous inequality by h we get

d dt h ≤ CEū h + F (3.4.24)
for some constant C > 0.

Deriving one time (Hyp) with respect to x, multiplying by ∂ x h and thank to

R ∂ x ( ū∂ x h)∂ x hdx = - 1 2 R ū(∂ x h) 2 dx
We obtain the estimate

d dt ∂ x h ≤ CEū ∂ x h + ∂ x F (3.4.25)
By combining the inequalities (3.4.24) and (3.4.25) we have for some constant C > 0:

d dt h 1 ≤ CEū h 1 + F 1 (3.4.26)
The Gronwall lemma gives (3.4.23).

Existence. We use the characteristic method: we define

     d ds χ(s, x, t) = u(s, χ(s, x, t)) χ(t 0 , x, t) = x 0 ∈ R
where χ(s, x, t) is a characteristic curve. Then the solution of (Hyp) takes the form:

∀t ∈ [0, T ], ∀y ∈ R, h(t, y) = h 0 (χ(0, y, t)) + t 0 F (τ, y, t))dτ
which gives the smoothness of h in C(0, T ;

H 1 (R)) ∩ C 1 (0, T ; L 2 (R)). Proposition 3.4.3. Let T > 0 such that F ∈ C(0, T ; H 2 (R)), ū ∈ L ∞ (0, T ; H 2 (R) and h 0 ∈ H 2 (R) Then 1.
The problem (Hyp) has an unique strong solution h such that:

h ∈ C(0, t; H 2 (R)) ∩ C 1 (0, T ; H 1 (R)) (3.4.27)
2. For all t ∈ [0, T ] we have the energy inequality Proof. Uniqueness. We differentiate (Hyp) two times with respect to x, we multiply by ∂ xx h and we integrate in space:

h(t, .) 2 ≤ e C 2 Eūt h 0 2 + t 0 e -C 2 Eūt F (τ, .
1 2 d dt ∂ xx h 2 = - R ∂ xx ( ū∂ x h)∂ xx h + R ∂ xx F ∂ xx h
by integration by parts we have: Existence. We differentiate (Hyp) with respect to x. We define φ :

R u∂ xxx h∂ xx h = - 1 2 R u∂ x ((∂ xx h) 2 ) = + 1 2 R ∂ x (u)(∂ xx h) 2 Since d dt ∂ xx h ≤ CEū h 2 + ∂ xx F (3.
= ∂ x h, then it verifies    ∂ t φ + ū∂ x φ = ∂ x F -∂ x ( ū)φ φ(0, x) = ∂ x h 0 (x) ∈ H 1 (R)
We follow an iterative technique to solve the previous problem with the initial condition

φ 0 (t, x) = ∂ x h 0 (x), ∀(t, x) ∈ [0, T ] × R Define, for j ≥ 1, φ (j) with    ∂ t φ (j) + ū∂ x φ (j) = ∂ x F -∂ x ( ū)φ (j-1) φ (j) (0, x) = ∂ x h 0 (x) ∀x ∈ R) Since ∂ x F -∂ x ( ū)φ (j-1) 1 ≤ F 2 +CEū φ (j-1)
1 thus, we have φ (j) ∈ C(0, T ; H 1 R)) according to the lemma 4.3. To prove the convergence of φ (j) j to ∂ x h, we write

∂ t φ (j+1) -φ (j) = ∂ x ( ū) φ (j) -φ (j-1)
if we use j times the energy estimate (3.4.28), we obtain

φ (j+1) -φ (j) 1 ≤ e C 2 Eūt t 0 e -C 2 Eūτ C 2 Eū φ (j) -φ (j-1) 1 dτ ≤ ... ≤ e C 2 Eūt (C 2 Eūt) j j! 2 ∂ x h 0 1 + t 0 e C 2 Eūτ ∂ x F (τ ) 1 dτ
The right term tends to zero as j goes to +∞. This gives the convergence of φ (j) j to ∂ x h in H 1 , and then the H 2 -regularity of h.

Proposition 3.4.4. Let S ∈ C(0, T ; H 1 (R)), h ∈ C(0, T ; H 3 (R)), and ū ∈ L ∞ (0, T ; H 2 (R), then 1. the system    ∂ t U -ν∂ xx U = S ∂ t h + ū∂ x h = F
has an unique strong solution (U,h). which belong to

U ∈ C(0, T ; H 2 (R)) ∩ C 1 (0, T ; L 2 (R)) ∩ L 2 (0, T ; H 3 (R)) h ∈ C(0, T ; H 2 (R)) ∩ C 1 (0, T ; H 1 (R)) (3
(U, h)(t, .) 2_2 ≤ Ke C(1+E ū, h) 2 t (U 0 , h 0 ) 2_2 + t 0 S(τ, .) 2 1 dτ 1/2 and t 0 U (τ, .) 2 3 dτ 1/2 ≤ Ke C(1+E ū, h) 2 t (U 0 , h 0 ) 2_2 + t 0 S(τ, .) 2 1 dτ 1/2 (3.4.31)
Proof. Uniqueness. According to the assumptions, the source term F of the hyperbolic equation belongs to

C(0, T ; H 1 (R)) ∩ L 2 (0, T ; H 2 (R))
and we observe that

F (τ, .) 2 ≤ C B (1 + E ū, h) U ((τ, .) 3 (3.4.32)
for some constant C B depending of z B . The Cauchy Schwartz inequality gives: (3.4.36). The uniqueness of the solution is proved by the priori estimate (3.4.31). Existence. By the proposition 3.4.1 S ∈ C(0, T ; H 1 (R)) and F ∈ H 1 (R). The proposition 3.4.2 gives the existence and uniqueness of U such that

t 0 e -C(k)E ū, hτ F (τ, .) k dτ ≤ e C 4 (1+E
U ∈ C(0, T ; H 2 (R)) ∩ C 1 (0, T ; L 2 (R)) ∩ L 2 (0, T ; H 3 (R)) Since F ∈ C(0, T ; H 1 (R)) ∩ C 1 (0, T ; L 2 (R)) ∩ L 2 (0, T ; H 2 (R))
and by the lemma 3.4.3 and the proposition 3.4.3, we obtain the existence and the uniqueness of h which satisfies:

h ∈ C(0, T ; H 1 (R)) ∩ C 1 (0, T ; L 2 (R))
To get more regularity in space, we derive the problem with respect to x, and

         ∂ t (∂ x U ) -ν∂ xx (∂ x U ) = ∂ x S ∂ t (∂ x h) + ū∂ x (∂ x h) = ∂ x ( F ) -∂ x ( ū)∂ x (h) (∂ x U 0 , ∂ x h 0 ) ∈ H 1 (R)
We can follow the same process as in the latest part of Proposition 3.4.3 to complete the proof.

Lemma 3.4.4. Let S ∈ C(0, T ; L 2 (R)), h ∈ C(0, T ; H 1 (R)), and ū ∈ L ∞ (0, T ; H 1 (R), then 1. The system

   ∂ t U -ν∂ xx U = S ∂ t h + ū∂ x h = F
has an unique strong solution (U,h). which belong to

U ∈ C(0, T ; H 1 (R)) ∩ C 1 (0, T ; L 2 (R)) h ∈ C(0, T ; H 1 (R)) (3.4.

37)

2. There exists C, K > 0 such that: for any t ∈ [0,T] Proof. The proof of this lemma is similar to the one of the proposition 3.4.4.

(U, h)(t, .) 1_1 ≤ Ke C(1+E
For proving the existence of solution of the non-linear system (3.4.12), we construct a sequence which is solution of a linear problem and converges to the strong solution of the non linear problem. This complete the proof of the theorem (3.4.1).

Picard's iterative scheme

We define a recursive sequence (U (j) , h (j) ) = (u

(j) 1 , ..., u (j)
N , h (j) ) j∈N as follows:

• ∀(t, x) ∈ [0, T ] × R, U (0) , h (0) (t, x) = (U 0 , h 0 )(x) ∈ H 2 (R) × H 3 (R) • ∀j ∈ N, (U (j+1) , h (j+1) ) is a solution of:          ∂ t U (j+1) -ν∂ xx U (j+1) = S(j) on [0, T ] × R ∂h (j+1) -ū(j) ∂ x h (j+1) = F (j,j+1) on [0, T ] × R (U (j) , h (j) )(0, .) = (U (0) , h (0) )(.) on R (3.4.39)
where for any j ∈ N:

   S (j) = S(U (j) , h (j) , ∂ x U (j) , ∂ x h (j) , ∂ xx h (j) ), F (j,j+1) = -G 1/2 -∂ x (ū (j+1) )h (j)
We also define

   E 0 = 2 (U 0 , h 0 ) 2,3 η 0 = 1 2 inf x∈R h 0 (x) Proposition 3.4.5. Let (h 0 , U 0 ) ∈ H 3 (R) × H 2 (R).
Then there exists a small T > 0 such that • The sequence (U (j) , h (j) ) j∈N is well defined and satisfies, for any t ∈ [0, T ] and any j ∈ N:

U (j) ∈ C (0, T, H 2 (R)) ∩ C 1 (0, t; L 2 (R)) ∩ L 2 (0, T ; H 3 (R)) h (j) ∈ C(0, T ; H 2 (R)) ∩ C 1 (0, T ; H 1 (R)) (3.4.40)
• ∀(t, x) ∈ [0, T ] × R and ∀j ∈ N, we have:

(U (j) , h (j) )(t, .) 2_2 ≤ E 0 t 0 U (j) (τ, .) 2 3 dτ 1/2 ≤ E 0 (3.4.41) h (j) (t, x) ≥ η 0 > 0 (3.4.42)
For the proof, we introduce the following regularization result (for more details see [START_REF] Matsumura | The initial value problem of the equations of motion of viscous and heat-conductive gases[END_REF]) Lemma 3.4.5. Let (U, h) ∈ C(0, T ; H 1 (R)) then There exist two positive constants C and C ′ such that

C ε 0 1 ≤ C U 2 (3.4.43) C ε 1 1 ≤ C ′ (h, U ) 2 (3.4.44)
Proof. (of proposition 3.4.5) Applying Proposition 3.4.4 on (U (0) , h (0) ), gives the existence (U (1) , h (1) ) ∈ C(0, t; H 2 (R)) for any t > 0. The characteristic formula applied to h (1) , yields, ∀t > 0:

h (1) (t, y) = h 0 (χ(0, y, t)) + t 0 F (0,1) (s, χ(s, y, t))ds

≥ 2η 0 + C(E 0 )t ≥ η 0
Which proves (3.4.42) for j = 1. We apply the inequality (3.4.31) in Proposition 3.4.4 to (U (1) , h (1) ), we get for any t ≤ T 1 :

(U (1) , h (1)

)(t, .) 2_2 ≤ Ke C(1+E ū, h) 2 t E 0 2 + t 0 S (0) (τ, .) 2 1 dτ 1/2 (3.4.45) and t 0 U (1) (τ, .) 2 3 dτ 1/2 ≤ Ke C(1+E ū, h) 2 t E 0 2 + t 0 S (0) (τ, .) 2 1 dτ 1/2 (3.4.46)
We use the Proposition 3.4.1 with S (0) , we obtain 102CHAPTER 3. SHAPE OPTIMIZATION OF FISHWAYS IN MULTILAYER FLOW BASED ON OPTI max (U (1) , h (1) )(t, .) 2_2 ,

t 0 U (1) (τ, .) 2 3 dτ 1/2 ≤ Ke C(1+E ū, h) 2 t E 0 2 + C(E 0 )(1 + E 0 + E 2 0 + E 3 0 ) + g z b 2 + 1 ρ p S 2 √ t We define for t ∈ R + ς(t) = Ke C(1+E ū, h) 2 t E 0 2 + C(E 0 )(1 + E 0 + E 2 0 + E 3 0 ) + g z b 2 + 1 ρ p S 2 √ t
The continuity of ς regarding to time t allows to find 0 < T 1 = T 1 (η 0 , E 0 , z b , p S ) such that (3.4.41) is satisfied for any t ≤ T 1 . We choose T := T 1

Next, we pass from j to j+1. If for any j ∈ N, (U (j) , h (j) ) satisfies (3.4.40), (3.4.41) and (3.4.42) for any t ≤ T 1 , Proposition 4.4 shows the existence of (U (j+1) , h (j+1) ).

We apply the inequality (3.4.31) to (U (j+1) , h (j+1) ), we get with similar manner, for any

t ≤ T 1 : (U (j+1) (t, .), h (j+1) )(t, .) 2_2 ≤ Ke C(1+E ū, h) 2 t E 0 2 + t 0 S (j) (τ, .) 2 1 dτ 1/2
Finally, we use the inequality revealed in Proposition 3.4.1 with S (j) , we obtain

(U (j+1) , h (j+1) )(t, .) 2_2 ≤ Ke C(1+E ū, h) 2 t E 0 2 + C(E 0 )(1 + E 0 + E 2 0 + E 3 0 ) + g z b 2 + 1 ρ p S 2 √ t
As the previous way, for T = T 1 , The estimation (3.4.41) is satisfied for any t ≤ T 1 . We choose T := T 1 . The inequality (3.4.28) for rank j+1 can be shown as the same manner of rank 1, this ends the proof with T := T 1 Now, it remains to prove the convergence of the sequence built above. In this order, we will proceed as the following way, for j > 1 the couple (U (j+1) -U (j) , h (j+1)h (j) ) satisfies

         (∂ t -ν∂ xx ) U (j+1) -U (j) = S (j) -S (j) ∂ t (h (j+1) -h (j) ) -ū(j) ∂ x (h (j+1) -h (j) ) = F (j,j+1) -F (j-1,j) -∂ x h (j) (ū (j) -ū(j-1) )
U (j+1) -U (j) , h (j+1)h (j) (0, .) = 0 (3.4.47) The source term of the second equation can be written as

F = F (j,j+1) -F (j-1,j) -∂ x h (j) (ū (j) -ū(j-1) ) = -h (j) (ū (j+1) -ū(j) ) -∂ x ū(j) (h (j) -h (j-1) ) -∂ x h (j) (ū (j) -ū(j-1) )
Using Lemma 3.4.1 yields to

F 1 ≤ CE 0 (U (j) -U (j-1) , h (j) -h (j-1) ) 1_1 +C B (1 + E 0 ) U (j+1) -U (j) 2 (3.4.48)
and

F 2 ≤ CE 0 (U (j) -U (j-1) , h (j) -h (j-1) ) 2_2 +C B (1 + E 0 ) U (j+1) -U (j) 3 (3.4.49)
With a similar manner as (3.4.17) we obtain

S (j) -S (j-1) ≤ C(1 + E 0 + E 2 0 + E 3 0 ) (U (j) -U (j-1) , h (j) -h (j-1) ) 1_1
Using (3.4.38) proved in the lemma 4.4, for any t ≤ T we have the following estimate

U (j+1) -U (j) , h (j+1) -h (j) (t, .) 1_1 ≤ C(E 0 )e C(1+E 0 ) 2 t t 0 U (j) -U (j-1) , h (j) -h (j-1) (τ, .) 2 1_1 1/2
We apply the previous inequality j times to get

U (j+1) -U (j) , h (j+1) -h (j) (t, .) 1_1 ≤ C(E 0 ) j e Kt sup 0≤t≤T U (1) -U (0) , h (1) -h (0) 2 1_1 1/2 1 K j j! with K = C(1 + E 0 ) 2 .
Which gives the existence of the subsequence (U (j) , h (j) ) j such as

(U (j) , h (j) ) -→ j→∞ (U,h) strongly in C(0, t; H 1 (R))
Moreover, Proposition 4.5 gives, up to subsequence, the convergence:

U (j) ⇀ j→∞ U weakly in L 2 (0, T ; H 3 (R)),
while for every fixed t ≤ T :

(U (j) , h (j) ) ⇀ j→∞ (U,h) weakly in H 2 (R)
The solution (U,h) for the system (3.4.12) belongs to C(0, T ; Finally, it remains to prove (U, h) ∈ C(0, T ; H 2 (R)). By regularizing: we consider (U ε , h ε ) = (ρ ε * U, ρ ε * h), where ρ ε * is the Friedrich's mollifier with respect to the space variable. If we apply ρ ε * to system (3.4.12) we get:

H 1 (R)) ∩ L ∞ (0, T ; H 2 (R)), satisfying ∀t, x ∈ [0, T ] × R 104CHAPTER
         ∂ t U ε -ν∂ xx U ε = S ε + C ε 0 ∂ t h ε -ūε ∂ x h ε = F ε + C ε 1 (U ε , h ε )(0, .) = (ρ ε * U 0 , ρ ε * h 0 ) ∈ C ∞ (3.4.50)
where

S ε = ρ ε * S F ε = ρ ε * F and    C ε 0 = (∂ t -ν∂ xx )(U ε ) -ρ ε * (∂ t U -ν∂ xx U ) C ε 1 = (∂ t h ε -∂ x (h ε ūε )) -ρ ε * (∂ t h -∂ x (hū))
The lemma 3.4.5 gives as ε goes to zero:

   C ε 0 , C ε 1 -→ 0, (U ε , h ε ) -→ (U, h).
Therefore, at the uniform limit we have (U,h) ∈ C(0, T ; H 2 (R)). The energy estimate gives the uniqueness and this end the proof of theorem 3.4.1.

The study of the incompressible Navier-stokes equations inspires an approach which we have to use to establish a multilayer system, and show the existence and uniqueness of its solution. In the following section, we are interested in design optimisation of vertical slots fishway. For this aim, we will use the hydrostatic Navier-Stokes equations, and deduce the associated multilayer system.

At the first step, we consider the height of water as a subdivision of layers α, and find the height and the horizontal velocity of water in each layer α. The next step consists to find an optimal shape of fishway corresponding to a velocity close to a given target velocity. Finally, it remains to derive the adjoint system from the multilayer state system in each layer, associated to the hydrostatic Navier-Stokes equations.

3-D Multilayer system and fishway shape optimization

In order to treat an optimal problem requiring a 2-D horizontal velocity, we are oriented towards a generalization of the previous multilayer method to the 3-D hydrostatic Navier-Stokes equations.

We suppose that u(t, x, z) = (u, v, w) is the velocity vector and let U (t, x, z) = (u, v) be the horizontal velocity. The hydrostatic model of Navier-Stokes system writes

               ∇.u = 0 ∂U ∂t + ∇ x .(U ⊗ U ) + ∂(U w) ∂z + ∇ x p = µ ∂ 2 U ∂z 2 ∂p ∂z = -g (3.5.1)
with the boundary conditions

w(t, x, z b ) = 0 µ ∂U ∂z (t, x, z b ) = kU (t, x, y, z b ) ∂U ∂z (t, x, h) = 0 p(t, x, h) = 0 (3.5.2)
and the kinematic boundary condition

∂h ∂t + U (t, x, h)∇ x h -w(t, x, h) = 0 (3.5.3)
The combination of the third equation of the system (3.5.1) and the fourth boundary condition of (3.5.2) gives the expression of the pressure p(t,x,z)

p(t, x, z) = g(h -z) (3.5.4)
The following subsection describes the technique for extracting the 3-D multilayer system from the system (3.5.1).

3-D Multilayer system

The water height is divided in the vertical direction z into a set of layers. In each layer α, for α = 1, ..., N , let us denote by h α the water height, where

h α = z α+1/2 -z α-1/2 , with z α+1/2 = z b + α β=1 h β 106CHAPTER 3.

SHAPE OPTIMIZATION OF FISHWAYS IN MULTILAYER FLOW BASED ON OPTI

with z b is the bathymetry of the bottom. and we define the averaged horizontal velocity u α as

u α (t, x) = 1 h α z α+1/2 z α-1/2 U (t, x, z)dz
We introduce an indicator function θ α (t, x, z) in each layer

θ α (t, x, z) =    1 if z α-1/2 ≤ z ≤ z α+1/2 0 otherwise , for α = 1,..., N which verifies ∂θ α ∂t + ∇ x .(θ α U ) + ∂θ α w ∂z = 0 (3.5.5)
Integrating the equation (3.5.5) in the vertical direction z, taking into account the condition (3.5.3) in each layer α, and using of the definition of u α (t, x) yields

∂h α ∂t + ∇ x .(h α u α ) = 0
Now, we integrate the equation (3.5.4) in each layer α, the form of hydrostatic pressure corresponds to

z α+1/2 z α-1/2 ∇ x pdz = z α+1/2 z α-1/2 g∇ x hdz = g∇ x h z α+1/2 z α-1/2 dz = gh α ∇ x h (3.5.6)
If we integrate the second equation of (3.5.1) in the vertical direction z, and take into account the kinematic boundary condition (3.5.3), we get for α = 1, ..., N ∂ ∂t

z α+1/2 z α-1/2 U dz+ ∇ x z α+1/2 z α-1/2 U ⊗ U dz + gh α ∇ x h = -gh α ∇ x z b +µ ∂U ∂z (t, x, z α+1/2 ) -µ ∂U ∂z (t, x, z α-1/2 ) (3.5.7)
We apply a finite difference method in the vertical direction to the term

µ ∂U ∂z (t, x, z α+1/2 ) -µ ∂U ∂z (t, x, z α-1/2 ),
we obtain for a horizontal velocity u α and water hight h α in each layer α, the following multilayer model of the system (3.5.1)

                 ∂h α ∂t +∇ x .(h α u α ) = 0 ∂h α u α ∂t +∇ x .(h α u α ⊗ u α ) + gh α ∇ x h = -gh α ∇ x z b -k α u α +2ν α u α+1 -u α h α+1 + h α -2ν α-1 u α -u α-1 h α + h α-1 f or α = 1, ..., N (3.5.8)
with

k α =    k if α = 1 0 if α = 1 , ν α =          0 if α = 0 µ if α = 1, ..., N 0 if α = N
We can write the previous system in terms of conservative components in the left hand side as follow:

                 ∂h α ∂t +∇ x .(h α u α ) = 0 ∂h α u α ∂t +∇ x .(h α u α ⊗ u α ) + g 2 ∇ x (h α h) = g 2 h 2 ∇ x h α h -gh α ∇ x z b -k α u α +2ν α u α+1 -u α h α+1 + h α -2ν α-1 u α -u α-1 h α + h α-1 f or α = 1, ..., N (3.5.9)
In the following, we give a definition for the shape optimization problem. We also define the geometry of the fishway to be optimized and give the system which governs the channel.

Optimal problem

In this section we are interesting in fishway structures. They are composed of succession of pools. There are also two transition pools, one in the beginning and the second at the end of structure. Two baffles are built in each pool and are vertical to the lateral channel boundary. The fishway is built with a slop relative to the ground (Figure 3.6)

                         ∂h α ∂t + ∇ x .q α = 0 in Ω α × [0, T ] ∂q α ∂t + ∇ x .( q α h α ⊗ q α ) + g 2 ∇ x (h α h) = g 2 h 2 ∇ x h α h -gh α ∇ x z b -k α q α h α +2ν α h α q α+1 -h α+1 q α h 2 α+1 h α + h 2 α h α+1 -2ν α-1 h α-1 q α -h α q α-1 h 2 α h α-1 + h 2 α-1 h α in Ω α × [0, T ] f or α = 1, ..., N (3.5.10)
We denote by γ 0 the lateral channel boundary, and by γ 1 α and γ 2 α the inflow/outflow boundary in each layer α respectively. Then the initial and boundary conditions are written as:

                   h α (0) = h 0 , q α (0) = q 0 in Ω α q α .n = 0, curl q α h α = 0 in γ 0 × [0, T ] q α = q 1 n in γ 1 α × [0, T ] h α = h 2 in γ 2 α × [0, T ] (3.5.11)
We take into account that the global geometry of each pool depends only of the two points a and b at the end of baffles (see figure 4.2). The position of these points change along flow for controlling the water velocity and permitting to maximum of fishes species to across the dam construction. In this order, two constraints are written:

1 4 1.213 ≤ y 1 , y 3 ≤ 3 4 1.213 0 ≤ y 2 , y 4 ≤ 1 4 0.97 (3.5.12)
Other constraints are adding for assuming a comfortable passage of a maximum number of fishes. These constraints are written as: The optimization problem consists in the minimization of a functional J(.) ∈ R, also called objective function depending on the variables a and b (Figure 4.2). The purpose of our work is to establish an optimal fishway structure and obtaining a velocity close to a target velocity named v in each layer α depended of fishes swimming ability. The velocity v is given by:

y 3 -y 1 ≥ d 1 = 0.1
v(x 1 , x 2 ) = (c, 0) if x 2 ≤ 1 3 0.97 (0, 0) otherwise (3.5.14)
and the objective function in each layer is defined as:

j α = 1 2 T 0 Ωα q α h α -v 2 dxdt + σ 2 T 0 Ωα | curl( q α h α ) | 2 dxdt (3.5.15)
with σ ≥ 0 is the vorticity parameter, and q α = h α u α where (h α , q α ) is solution of the multilayer system (3.5.10) with the initial and boundary conditions (3.5.11).

The global objective function to be minimized takes the form

J = N α=1 j α (3.5.16)
In the following subsection, we derive an adjoint system related to the state system (3.5.10) and the objective function (3.5.15).

Adjoint system

We consider (p, r) two test functions space in L ∞ (0, T ; H 2 (Ω)) 2 . The variational formulation of the state system (3.5.10) in each layer α for α = 1, ..., N is given by

110CHAPTER 3. SHAPE OPTIMIZATION OF FISHWAYS IN MULTILAYER FLOW BASED ON OPTI F ind (h α , q α ) ∈ L ∞ (0, T ; H 2 (Ω)) 2 such that A α (Ω α , h α , q α ; p, r) = L α (Ω α ; p, r) ∀(p, r) ∈ L ∞ (0, T ; H 2 (Ω)) 2
where

A α (Ω α , h α , q α ; p, r) = T 0 Ωα ∂h α ∂t + ∇.q α .p + T 0 Ωα ∂q α ∂t + ∇.( q α h α ⊗ q α ) + g 2 ∇(h α h) .r - T 0 Ωα g 2 h 2 ∇ h α h -gh α ∇z b -k α q α h α .r - T 0 Ωα 2ν α h α q α+1 -h α+1 q α h 2 α+1 h α + h 2 α h α+1 -2ν α-1 h α-1 q α -h α q α-1 h 2 α h α-1 + h 2 α-1 h α .r
We rewrite the cost function in each layer as J α (Ω α , h α , q α ) = j α (Ω), such that

J α (Ω α , h α , q α ) = 1 2 T 0 Ωα q α h α -v 2 dtdx + σ 2 T 0 Ωα | curl( q α h α ) | 2 dtdx (3.
                                                             - ∂p ∂t + 1 h 2 α (q α .∇)r.q α + g∇h.r + g∇z b .r -k α q α h 2 α .r -2ν α h 3 α+1 q α -h 2 α h α+1 q α+1 + 2h α h 2 α+1 q α (h 2 α+1 h α + h 2 α h α+1 ) 2
.r

+ 2ν α-1 2h 2 α h α-1 q α-1 -2h α h 2 α-1 q α -h α h 2 α-1 q α-1 -h 3 α-1 q α (h 2 α h α-1 + h 2 α-1 h α ) 2 .r = - q α h α -v q α h 2 α -σcurl(curl( q α h α )). q α h 2 α in Ω α × [0, T ] - ∂r ∂t -∇p - 1 h α (q α .∇)r - 1 h α (∇r) t q α + k α r h α + 2ν α h α+1 r h 2 α+1 h α + h 2 α h α+1 + 2ν α-1 h α-1 r h 2 α h α-1 + h 2 α-1 h α = 1 h α ( q α h α -v) + σ 1 h α curl(curl( q α h α )) in Ω α × [0, T ] (3.5.18)
with the final and boundary conditions

p(T ) = 0, r(T ) = 0 in Ω α r.n = 0 in γ 0 × [0, T ] - q 1 h 2 α .r.n = 0 in γ 1 × [0, T ] (p + 1 h 2 (q α .r))n + 1 h 2 (q α .n).r - σ h 2 curl( q α h 2 )τ = 0 in γ 2 α × [0, T ] (3.5.19)
where (h α , q α ) is solution of (3.5.10)-(3.5.11)

Proof. We define a Lagrangian function as the following way:

L ag (Ω α , h α , q α ; p, r) = J α (Ω α , h α , q α ) -A α (Ω α , h α , q α ; p, r) + L α (Ω α ; p, r)
By applying an integration by parts on the function A α (Ω α , h α , q α ; p, r), deriving with respect to h α the function A α and J α , and using the adjoint state of L ag (see [4])

∂ ∂h α L ag (Ω α , h α , q α ; p, r). hα = 0
for any arbitrary direction hα , to obtain the following system: in

Ω α × [0, T ] - ∂p ∂t + 1 h 2 α (q α .∇)r.q α + g∇h.r + g∇z b .r -k α q α h 2 α .r -2ν α h 3 α+1 q α -h 2 α h α+1 q α+1 + 2h α h 2 α+1 q α (h 2 α+1 h α + h 2 α h α+1 ) 2
.r

+ 2ν α-1 2h 2 α h α-1 q α-1 -2h α h 2 α-1 q α -h α h 2 α-1 q α-1 -h 3 α-1 q α (h 2 α h α-1 + h 2 α-1 h α ) 2 .r = - q α h α -v q α h 2 α -σcurl(curl( q α h α )). q α h 2 α
with final and boundary conditions

p(T ) = 0 in Ω α r.n = 0 in γ 0 × [0, T ] - q 1 h 2 α .r.n = 0 in γ 1 × [0, T ]
Next, we derive with respect to u α the functions A α and J α , then we use the adjoint state ∂ ∂q α L ag (Ω α , h α , q α ; p, r).q α = 0 for any arbitrary direction qα , to get: in

Ω α × [0, T ] 112CHAPTER 3. SHAPE OPTIMIZATION OF FISHWAYS IN MULTILAYER FLOW BASED ON OPTI - ∂r ∂t -∇p - 1 h α (q α .∇)r - 1 h α (∇r) t q α + k α r h α + 2ν α h α+1 r h 2 α+1 h α + h 2 α h α+1 + 2ν α-1 h α-1 r h 2 α h α-1 + h 2 α-1 h α = 1 h α ( q α h α -v) + σ 1 h α curl(curl( q α h α ))
with final and boundary conditions:

r(T ) = 0 in Ω α r.n = 0 in γ 0 × [0, T ] (p + 1 h 2 (q α .r))n + 1 h 2 (q α .n).r - σ h 2 curl( q α h 2 )τ = 0 in γ 2 α × [0, T ]
The following subsection gives results about the domain derivative of the objective function (3.5.15) in each layer α in terms of adjoint and state variables.

Domain derivative

The study of the cost function will be done in each layer α. Now, let Ω 0 α be a fixed domain, and we suppose that it is a bounded open subset of R 2 with Lipschitz boundary. We denote by Lip(Ω 0 α ; R 2 ) the Lipschitz applications domain, which is defined by

Lip(Ω 0 α ; R 2 ) = {φ : Ω 0 α → R 2 / ∃k ≥ 0 such that φ(x) -φ(y) ≤ k x -y , ∀(x, y) ∈ Ω 0 α × Ω 0 α }
We denote by O 0 α the subset of bi-Lipschitz homeomorphisms of Ω 0 α . It is an open subset of Lip(Ω 0 α ; R 2 ) and is given by

O 0 α = {ψ : Ω 0 α → ψ(Ω 0 α ), bijective / ψ ∈ Lip(Ω 0 α ; R 2 ), ψ -1 ∈ Lip(ψ(Ω 0 α ); R 2 )}
Then, the admissible domain denoted by X 0 α is a set of domains Ω α which are bound by bijective Lipschitz functions to the fixed domain Ω 0 α . It is given as the follow way

X 0 α = Ω α = ψ(Ω 0 α ) / ψ ∈ O 0 α (3.5.20) Theorem 3.5.2. Let V ∈ Lip(Ω α ; R 2 ), then the domain derivative of j α at Ω α ∈ X 0 α takes the form ∂ ∂Ω α j α (Ω α ).V = ∂ ∂Ω α J α (Ω α ; h α , q α ).V - ∂ ∂Ω α A α (Ω α ; h α , q α ; p, r).V + ∂ ∂Ω α L α (Ω α ; p, r).V (3.5.21)
where (h α , q α ) is solution of the state system (3.5.10) with initial and boundary conditions (3.5.11), and (p, r) is solution of the adjoint system (3.5.18) with final and boundary conditions (3.5.19).

In the following, we give the computation details for each derivative term cited in the previous theorem.

Let

Ω α ∈ X 0 α , such that Ω α = ψ(Ω 0 α ) with ψ ∈ O 0 α . We consider F ∈ Lip(Ω α ; R 2 ) a bi-Lipschitz homeomorphism such that ω α = F (Ω α ) ∈ X 0 α . Let V ∈ Lip(Ω α ; R 2 ) such that V = F -I.
We define the function d by d : Ω α ∈ X 0 α → d(ω α ) ∈ R, and we introduce a transported function called d as the follow way

d : F ∈ Lip(Ω α ; R 2 ) → d(F ) = d(F (Ω α )) = d(ω α ) ∈ R.
Thus, the domain derivative of d at a given Ω α ∈ X 0 α is defined by

∂ ∂Ω α d(Ω α ).V = ∂ ∂F d(I).V , f or V ∈ Lip(Ω α ; R 2 ) (3.5.22)
The computation of the domain derivative of j α is done by introducing the transported functions of A α , L α , and J α . For this aim, we define the function A α as a sum of a number of functions as follow

A α (Ω α , h α , q α ; p, r) = T 0 Ωα ∂h α ∂t p A 1 α (Ωα,hα,q α ;p,r) + T 0 Ωα (∇ x .q α )p
A 2 α (Ωα,hα,q α ;p,r)

+ T 0 Ωα ∂q α ∂t .r
A 3 α (Ωα,hα,q α ;p,r)

+ T 0 Ωα ∇ x .( q α h α ⊗ q α ).r A 4 α (Ωα,hα,q α ;p,r) + T 0 Ωα g 2 ∇ x (h α h).r A 5 α (Ωα,hα,q α ;p,r) - T 0 Ωα g 2 h 2 ∇ x h α h .r
A 6 α (Ωα,hα,q α ;p,r)

+ T 0 Ωα gh α ∇ x z b .r A 7
α (Ωα,hα,q α ;p,r)

+ T 0 Ωα k α q α h α .r A 8 α (Ωα,hα,q α ;p,r) - T 0 Ωα 2ν α h α q α+1 -h α+1 q α h 2 α+1 h α + h 2 α h α+1 .r
A 9 α (Ωα,hα,q α ;p,r)

+ T 0 Ωα 2ν α-1 h α-1 q α -h α q α-1 h 2 α h α-1 + h 2 α-1 h α .r
A 10 α (Ωα,hα,q α ;p,r) 114CHAPTER

SHAPE OPTIMIZATION OF FISHWAYS IN MULTILAYER FLOW BASED ON OPTI

We kept J α as

J α (Ω α , h α , q α ) = 1 2 T 0 Ωα q α h α -v 2 J 1 α (Ωα,hα,q α ) + σ 2 T 0 Ωα | curl( q α h α ) | 2 J 2 α (Ωα,hα,q α )
Next, for computing the domain derivative of each function above, we introduce its transported function. In particular for the first function

A 1 α (Ω α , h α , q α ; p, r) = T 0 Ωα ∂h α ∂t p,
we introduce the transported function

A 1 α (F , h α , q α ; p, r) = A 1 α (F (Ω α ), h α • F -1 , q α • F -1 ; p • F -1 , r • F -1 ) = A 1 α (ω α , h α , q α ; p, r) = T 0 Ωα ∂h α ∂t p | det(∇ x F ) | As it is known, F →| det(∇ x F ) | is a differentiable application. Its derivative writes ∂ ∂F | det(∇ x F ) | (I).V = ∇ x .V , f or V ∈ Lip(Ω α ; R 2 ) Hence, for V ∈ Lip(Ω α ; R 2 ) ∂ ∂Ω α A 1 α (Ω α , h α , q α ; p, r).V = ∂ ∂F A 1 α (I, h α , q α ; p, r).V = T 0 Ωα ∂h α ∂t p(∇ x .V )
For the second function A 2 α (Ω α , h α , q α ; p, r) , we consider the following transported function

A 2 α (F , h α , q α ; p, r) = A 2 α (F (Ω α ), h α • F -1 , q α • F -1 ; p • F -1 , r • F -1 ) = A 2 α (ω α , h α , q α ; p, r) = T 0 Ωα (∇ x F -1 • F ) t : ∇ x (q α )p | det(∇ x F ) |
where the inner product "X:Y" is expressed as

X : Y = 2 k,l=1 X kl Y kl .
Taking into account that the application F → ∇ x F -1 • F is differentiable and the derivative is written as

∂ ∂F (∇ x F -1 • F )(I).V = -∇ x .V , f or V ∈ Lip(Ω α ; R 2 ) Then, for V ∈ Lip(Ω α ; R 2 ) ∂ ∂Ω α A 2 α (Ω α , h α , q α ; p, r).V = ∂ ∂F A 2 α (I, h α , q α ; p, r).V = T 0 Ωα (∇ x .q α )p(∇ x .V ) - T 0 Ωα (∇ x V ) t : (∇ x q α )p
For the third function A 3 α (Ω α , h α , q α ; p, r) , we define the transported function

A 3 α (F , h α , q α ; p, r) = A 3 α (F (Ω α ), h α • F -1 , q α • F -1 ; p • F -1 , r • F -1 ) = A 3 α (ω α , h α , q α ; p, r) = T 0 Ωα ∂q α ∂t .r | det(∇ x F ) | Consequently, for V ∈ Lip(Ω α ; R 2 ) ∂ ∂Ω α A 3 α (Ω α , h α , q α ; p, r).V = ∂ ∂F A 3 α (I, h α , q α ; p, r).V = T 0 Ωα ∂q α ∂t .r(∇ x .V )
For the function A 4 α (Ω α , h α , q α ; p, r) , the associated transported function

A 4 α (F , h α , q α ; p, r) = A 4 α (F (Ω α ), h α • F -1 , q α • F -1 ; p • F -1 , r • F -1 ) = A 4 α (ω α , h α , q α ; p, r) = T 0 Ωα (∇ x F -1 • F ) t : ∇ x ( q α h α ⊗ q α ).r | det(∇ x F ) |
As the same manner used for the function A 2 α (Ω α , h α , q α ; p, r) we obtain for

V ∈ Lip(Ω α ; R 2 ) 116CHAPTER 3. SHAPE OPTIMIZATION OF FISHWAYS IN MULTILAYER FLOW BASED ON OPTI ∂ ∂Ω α A 4 α (Ω α , h α , q α ; p, r).V = ∂ ∂F A 4 α (I, h α , q α ; p, r).V = T 0 Ωα (∇ x .q α ) q α h α .r(∇ x V ) - T 0 Ωα (∇ x V ) t : (∇ x q α ) q α h α .r + T 0 Ωα (q α .∇ x ) q α h α .r(∇ x V ) - T 0 Ωα (∇ x V q α .∇ x ) q α h α .r
Finally, we get the domain derivative of the other functions for

V ∈ Lip(Ω α ; R 2 ) ∂ ∂Ω α A 5 α (Ω α , h α , q α ; p, r).V = T 0 Ωα g 2 ∇ x (h α h).r(∇ x .V ) - T 0 Ωα g 2 (∇ x V ) t ∇ x (h α h).r ∂ ∂Ω α A 6 α (Ω α , h α , q α ; p, r).V = T 0 Ωα g 2 h 2 ∇ x ( h α h ).r(∇ x .V ) - T 0 Ωα g 2 h 2 (∇ x V ) t ∇ x ( h α h ).r ∂ ∂Ω α A 7 α (Ω α , h α , q α ; p, r).V = T 0 Ωα gh α ∇ x z b .r(∇ x .V ) ∂ ∂Ω α A 8 α (Ω α , h α , q α ; p, r).V = T 0 Ωα k α q α h α .r(∇ x .V ) ∂ ∂Ω α A 9 α (Ω α , h α , q α ; p, r).V = T 0 Ωα 2ν α h α q α+1 -h α+1 q α h 2 α+1 h α + h 2 α h α+1 .r(∇ x .V ) and ∂ ∂Ω α A 10 α (Ω α , h α , q α ; p, r).V = T 0 Ωα 2ν α-1 h α-1 q α -h α q α-1 h 2 α h α-1 + h 2 α-1 h α .r(∇ x .V )
For J α (Ω α , h α , q α ; p, r) and L α (Ω α , h α , q α ; p, r), we follow the same approach and we get for

V ∈ Lip(Ω α ; R 2 ) ∂ ∂Ω α J 1 α (Ω α , h α , q α ).V = 1 2 T 0 Ωα | q α h α -v | 2 (∇ x .V ) ∂ ∂Ω α J 2 α (Ω α , h α , q α ).V = σ 2 T 0 Ωα | curl( q α h α ) | 2 (∇ x .V ) ∂ ∂Ω α L α (Ω α ; p, r).V = 0
By using all the previous results in (3.5.21), we obtain the formula for the domain derivative of the objective function j α at Ω α ∈ X 0 α in each layer α.

Numerical resolution

In this section, we formulate the multilayer system (3.5.10) into a conservative system with source term. let us denote X α = (h α , q α ) T with q α = (q α,x , q α,y ), then the system (3.5.10) can be written as

∂X α ∂t + ∇.F (h, X α ) = S p (h, X α ) + S z (X α ) + S ν (X α-1 , X α , X α+1 ) f or α = 1, ..., N (3.6.1) with F (h, X α ) =          q α,x q α,y q 2 α,x h α + g 2 h α h q α,x q α,y h α q α,x q α,y h α q 2 α,y h α + g 2 h α h          (3.6.2) S(h, X α ) =          0 g 2 h 2 ∂ x h α h g 2 h 2 ∂ y h α h          , S p (X α ) =      0 -gh α ∂ x z b -gh α ∂ y z b      (3.6.3) 118CHAPTER 3. SHAPE OPTIMIZATION OF FISHWAYS IN MULTILAYER FLOW BASED ON OPTI and S ν (X α-1 , X α , X α+1 ) =    0 -k α u α + 2ν α u α+1 -u α h α+1 + h α -2ν α-1 u α -u α-1 h α + h α-1    (3.6.4)
The approximation of the system of conservation laws (3.6.1) is based on a finite volume type dicscretization.

Finite volume method

Finite volume schemes for the shallow water systems consist in using an upwinding of the fluxes. The problem domain is first discretized into a set of triangular cells T i forming an unstructured computational mesh. Let ∆t be the constant time step and define t n = n∆t for n = 0, ..., N . At each discrete time t n , we note X i,n α the approximated solution value. By integrating the equation (3.6.1) on a triangle T i , we obtain

T i ∂ t X α + T i ∇.F (h, X α ) = T i S(h, X α+1 , X α , X α-1 ) (3.6.5)
where S(h, X α+1 , X α , X α-1 ) = S p (h, X α ) + S z (X α ) + S ν (X α-1 , X α , X α+1 ), and n i is the normal on the edges of triangle T i . By applying the divergence formula, the equation (3.6.5) writes as

T i ∂ t X α + ∂T i F (h, X α ).n i dΓ = T i S(h, X α+1 , X α , X α-1 ) (3.6.6)
Using a gauss formula for the second term of the left hand side to obtain

∂T i F (h, X α ).n i dΓ = 3 j=1 F ij .n ij .dl ij
The equation (3.6.6) takes the form

| T i | ∂ t X α + 3 j=1 F ij .n ij .dl ij =| T i | S (3.6.7)
where n ij is the normal on the edge T i /T j , F ij are the discrete fluxes on the interface T i /T j and dl ij is the length of the interface T i /T j . Therefore, we now have an equation for each cell i of the form

∂ t X α = - 1 | T i | 3 j=1 F ij .n ij .dl ij + S (3.6.8)
A finite difference approximation is adopted to obtain the scheme

X i,n+1 α = X i,n α - ∆t | T i | . 3 j=1 F ij .n ij .dl ij + ∆t.S (3.6.9)
The Roe solver is used to evaluate the term

3 j=1 F ij .n ij .dl ij .
In what follow, we will present two approaches. The first one does not need the gradient of the cost function trough its algorithm iterations. The second technique is called spectral projected gradient. It calls the gradient of the functional we want to minimize along the iterations. Finally we compare the results of each used technique.

A gradient free algorithm

The geometry of the vertical slot based on the use of guide elements to lead smooth hydraulic flow into the next slot.

The positioning of the guide elements was carried out at two different locations, y = (a, b) = (y 1 , y 2 , y 3 , y 4 ) (Figure 4.2), which configure the shape of the fish ladder. We formulate the shape optimization problem as a constrained problem depending on all linear constraints (3.5.12) and (3.5.13). We redefine the objective function (3.5.15) in the following way Φ 1 α : R 4 → R where Φ 1 α (y) = J α (Ω α (y)). The discrete approximation of the objective function in each layer α can be written as

Φ1 α (y) = ∆t 2 N n=1 e∈T i [ e u i,n α -- → v 2 + α e | curl(u i,n α ) | 2 ] (3.6.10)
Here u i,n α is the discrete horizontal velocity obtained by the finite volume method (3.6.9). Next, we collect all linear constraints (3.5.12) and (3.5.13) in a function For solving the non linear optimization problem, firstly we adopt a gradient free method named "Nelder-Mead" simplex technique. The Nelder-Mead technique, begins with an arbitrary simplex of 5 vertices y 1 , y 2 , ..., y 5 . it evaluates and order the cost function on these vertices Φ(y 1 ) ≤ Φ(y 2 ) ≤ ... ≤ Φ(y 5 ). The vertex associated to the maximal value is replaced with a new point y(ν) = (1 + ν)y *νy 5 , where y * is the centroid of the convex hull {y 1 , ..., y 4 }. The value of ν is chosen from this set of values: ν δ = -0.5, ν γ = 0.5, ν α = 1, ν β = 2. The choice of these values is determined according to the following algorithm Calculate and sort Φ(y 1 ), Φ(y 2 ), ..., Φ(y 5 ) While | Φ(y 5 ) -Φ(y 1 ) | is not sufficiently small, calculate y(ν β ) and To prevent stagnation at non-optimal point, a modification proposed by Kelley [3] is used. This technique consists to replace the current simplex by a smaller one.

φ 2 α : R 4 → R 10 φ 2 α (y 1 , y 2 , y 3 , y 4 ) = ( 1 4
Φ β = Φ(y(ν β )) then a) If Φ β ≤ Φ(y 1 ) then calculate Φ α = Φ(y(ν α )). If Φ β ≤ Φ α , replace

Spectral projected gradient

The second adopted approach to solve the shape optimization problem is consisting in the use of a gradient technique called Spectral projected gradient. This method use the projection into a closed and convex ω α subset of R 4 of all the points y ∈ R 4 satisfying (5.12)-(5.13) in each layer α.

We denote a 1 = a 3 = 1 4 1.213, b 1 = b 3 = 3 4 1.213, a 2 = a 4 = 0, b 2 = b 4 = 1 4
0.97. Then the admissible set ω α is defined as

ω α = {y = (y 1 , y 2 , y 3 , y 4 ) ∈ R 4 : a i ≤ y i ≤ b i , i = 1, ..., 4, y 3 -y 1 ≥ d 1 , y 2 -y 4 ≥ d 2 } (3.6.14)
The optimization problem can be reformulated as

min y∈ωα J α (y). ( 3
min y∈ωα 1 2 y -z 2 2 = min y∈ωα 1 2 z T z -z T y + 1 2 y T y (3.6.16) which is equivalent to                min (y 1 ,y 2 ,y 3 ,y 4 ) 4 i=1 1 2 z 2 i -z i y i + 1 2 y 2 i subject to a i ≤ y i ≤ b i , i = 1, ..., 4
y 3 -y 1 ≥ d 1 , y 2 -y 4 ≥ d 2 ⇔                                                  min (y 1 ,y 3 ) 1 2 (z 2 1 + z 2 3 ) -(z 1 y 1 + z 3 y 3 ) + 1 2 (y 2 1 + y 2 3 ) subject to a i ≤ y i ≤ b i , i = 1, 3 y 3 -y 1 ≥ d 1 ,
and

min (y 2 ,y 4 ) 1 2 (z 2 2 + z 2 4 ) -(z 2 y 2 + z 4 y 4 ) + 1 2 (y 2 2 + y 2 4 ) subject to a i ≤ y i ≤ b i , i = 2, 4 y 2 -y 4 ≥ d 2 ,
In order to solve the two quadratic problems, we use the Karush-Kuhn-Tucker (KKT) technique: Begin by the first quadratic problem and we choose a = a 1 = a 3 , and b = b 1 = b 3 , then from the five constraints, only three are kept. Finally the optimization problem is reformulated as

min (x 1 ,x2) c -(l 1 x 1 + l 2 x 2 ) + 1 2 (x 2 1 + x 2 2 ) subject to x 2 ≤ b, x 1 ≥ a, x 2 -x 1 ≥ d 1 ,
where ,a, b, d 1 , c, and l i , i=1,2, are real numbers. Let

A =    0 -1 1 0 -1 1    , b =    -b a d 1    ,
Then the linear constraints write as

Ax -v = b, v ≥ 0, where v = (v 1 , v 2 v 3 )
T is the slack variables. The cost function is strictly convex and quadratic in R 2 . Then the optimisation problem has an unique solution which satisfies the KKT conditions

v = Ax -b -l + x = A T w, v ≥ 0, w ≥ 0, v T w = 0,
with l = (l 1 , l 2 , l 3 ), and w = (w 1 , w 2 , w 3 ) is the vector of the Lagrange multipliers associated to the three constraints above.

The unique optimal solution satisfies

x = l + A T w
For computing the Lagrange multipliers w i , we solve the following LCP problem

v = (-b + Al) + AA T w v ≥ 0, w ≥ 0 v T w = 0
3.6.4 Numerical results

Transcritical flow over a bump

To validate the numerical method based on the use of the multilayer system treated in this work, we introduce the test transcritical flow over a bump which is proposed by Castro et al. [START_REF] Castro | On some fast well-balanced first order solvers for nonconservative systems[END_REF].

The domain is a channel of 21 m of length and 2 m of width with a bump of 5.75 m of length and it has a height of 0.2 m. For the boundary condition, we take a discharges of 2 m 3 /s at the inflow boundary , and a water height of 0.6 m in the outflow boundary. The vertical viscosity is 0.001 m 2 /s and the Strickler coefficient is 25. The numerical simulation for the three tests is done with the same physical and numerical conditions. All initial and boundary conditions are taken constants. h 0 = h α (x, 0) = 0.083m, q 0 = q α (x, 0) = (0; 0)m 2 s -1 , q 1 = -0.065/0.97 m 2 s -1 , and h 2 = 0.5 m. the target velocity is c = 0.8 m.s -1 . The vorticity parameter is α = 0. The penalty parameter is β = 500. In these experiments, we consider that only the bottom friction stress is applied associated to Chezy coefficient of 57.36. For fish passage with comfortable conditions, we put d 1 = 0.1 and d 2 = 0.05 (see comfort constraints (3.5.13)).

Slot Fishway ("I" shaped baffle-rectangular slots)

Numerical simulations are focused to study the rectangular baffles as shown in figure 4.2 (see [START_REF] Alvarez-Vazquez | An optimal shape problem related to the realistic design of river fishways[END_REF]). For initial random shape (figure 3.7 and 3.9), we observe circulation region near form baffles compared to the optimal shape the velocity is almost uniform and is close to the target velocity v. The turbulence flow is removed for optimal points positions a N M = (0.4640; 0. We consider that the baffles under study are obliques (Figure 3.12), instead of rectangular baffles as done in the first experiment. The stability constraints are written as

y 3 -y 1 ≥ d 1 = 0.1 y 2 -y 4 ≥ d 2 = 0.05 y 1 -y 5 ≥ d 3 = 1 2 0.0305 y 6 -y 2 ≥ d 4 = 1 2 0.0305 (3.6.18)
Finally, in each layer α, the objective function conserves the same writing. We collect all sixteen constraints defined in (3.6.17) and (3.6.18) in a function The associated penalty function takes the form

Φ 2 α : R 6 -→ R 16 Φ 2 α (y 1 , y 2 , y 3 , y 4 , y 5 , y 6 ) = ( 1 4 1.213 -y1, 1 4 1.213 -y 3 , 1 4 1.213 -y 5 , y 1 - 3 4 1.21, y 3 - 3 4 1
Φ α (y) = Φ1 α (y) + β 16 j=1 max{(Φ 2 α (y)) j , 0} (3.6.20)
For the spectral projected gradient, the matrix A and b are replaced by

A =      0 -1 0 0 0 1 1 0 -1 -1 1 0      , b = (- 3 4 1.213, 1 4
1.213, 1 2 0.0305, 0.1) T for the horizontal projection, and We observe the disappearance of the circulation area in the optimal structure (Figure 3.17

A =      0 0 -1 0 1 0 -1 0 1 1 -1 0      , b = (- 1 2 0.97, 0, 1 2

Conclusion

The multilayer model is adopted to avoid the expensive Navier-Stokes equations and is used in order to compute the free surface flow in vertical slot fishways. Design of effective fishways is becoming increasingly important. Numerical tests prove that the 2D multilayer Saint-Venant combined with an optimal shape design techniques is an alternative to 3D hydrostatic Navier-Stokes system. The numerical simulations evaluated the performance of various regular pool geometries. The study underlines the important influence of slot layout on flow characteristics. This study also assessed that the multi-slot design is suitable in terms of velocity and flow pattern.

-ν∆u ε + (u ε .∇)u ε + 1 2 (∇.u ε )u ε + a | u ε | α u ε + ∇p ε = f in Ω ∇.u ε + εp ε = 0 in Ω u ε = 0 on Γ 0 u ε = g on Γ 1 ν ∂u ε ∂n -p ε n = 0 on Γ 2 (4.1.2)
Along this chapter, we restrict the treatment to the case g = 0 for the system (4.1.1). We study the existence and uniqueness of a weak solution by using a Faedo-Galerkin method.

A penalized problem is introduced to derive an error priori estimates and show that the perturbed problem has a convergent solution as O(ε) where ε is the penalty parameter.

We study a penalized finite element method approximation to establish existence and uniqueness and to derive an error estimates for the approximated solution.

This kind of water flow is used to study a shape optimization problem of fishways. We look for finding an optimal shape of fishways by minimizing a cost function. Such optimal structure allow to a maximum number of fishes to cross to the river upstream. The construction of dams is a very old activity; the first known works date back to 5,000 years and are located in the Middle East. It has continued to grow ever since. These structure are used usually for agriculture, electricity, or for navigation, But they constitute impassable obstacles to the free movement of fish and may contribute to the extinction of fish populations as Salomon, Trout, and Eels. In order to resolve achieve this challenge, men think to built the hydraulic structure which permit to migratory species to cross to the river upstream. Fishways represent a comfortable way for migratory species to across through dams. Our objective in this chapter is to simply the fishes passage to their production or feeding areas and assure to provide a rest area between pools. In our study we are interested by the vertical slots fishways (see [START_REF] Rajaratnam | Hydraulics of vertical slot fishways[END_REF]).

Vertical slots fishways are used frequently in the world. The vertical slots provide a good energy dissipation and create a quiet areas between pool which allows to fishes to have some rest before crossing one pool to another. Such structure is designed with a slope to the ground. It permits to small species to pass the circulation zones in the pool with less efforts which is not the case for other types of fishways; as the pool and weir type (Clay [START_REF] Clay | Design of fishways and other Fish Facilities[END_REF]) and the Denil type (Katopodis et al. [START_REF] Katopodis | Denil Fishways of varying[END_REF]).

The chapter is organized as follows. The next section deals with the analysis of the Lemma 4.2.2. Assume that Ω is an arbitrary domain in R d (d=2, 3), and let f ∈ X ′ . Then 1. The vector V ∈ H 1 loc (Ω) satisfies (4.2.3) for all v ∈ C ∞ 0,div (Ω) if and only if there exists P ∈ L 2 loc (Ω) satisfying the identity

ν(∇V, ∇w) + ((V.∇)V, w) + (a|V | α V, w) = (P, ∇.w) + (f, w) (4.2.6)
for all w ∈ C ∞ 0,div (Ω).

2. If, Moreover, Ω is bounded and Lipschitzian and f

∈ X ′ , V ∈ (H 1 (Ω)) d (d=2,
F (w) := ν(∇V, ∇w) + ((V.∇)V, w) + (a|V | α V, w) -(f, w) (4.2.7)
is linear and bounded in w ∈ X and it becomes zero when w ∈ X and ∇.w = 0. Using the Corollary 3.5.1 in [START_REF] Galdi | An introduction to the Mathematical Theory of the Navier Stokes Equations[END_REF] and lemma 4.2.1, there exists P ∈ Y such that

F (w) = (P, ∇.w) (4.2.8)
for all w ∈ X, thus satisfying (4.2.6). If Ω is an arbitrary domain, we use Corollary 3.5.2 in [START_REF] Galdi | An introduction to the Mathematical Theory of the Navier Stokes Equations[END_REF] to deduce the existence of P ∈ L 2 loc (Ω) satisfying (4.2.8) ∀w ∈ C ∞ 0,div (Ω).

Lemma 4.2.3. Consider F: R m -→ R m be a continuous function such that for some K > 0

F (ξ).ξ > 0 for all ξ ∈ R m with | ξ |= K. Then there exists ξ 0 ∈ R m with | ξ 0 |≤ K such that F (ξ 0 ) = 0.
Proof. (of theorem (4.2.1)) Existence of the velocity field: We consider an orthonormal basis (w k ) in X and define the sequence (u m ) ∈ X satisfying

ν(∇u m , ∇w k ) + ((u m .∇)u m , w k ) + (a | u m | α u m , w k ) = (f, w k ) (4.2.9)
where for each m ∈ N

u m = m k=1 ξ km w k (4.2.10) 136CHAPTER 4

. ANALYSIS AND FINITE ELEMENT APPROXIMATION OF OPTIMAL CONTROL P

We multiply (4.2.9) by ξ km and sum with respect to k to obtain

ν ∇u m 2 +a u m α+2 L α+2 = (f, u m ) ≤ f X ′ ∇u m (4.2.11)
By Hölder inequality we get

ν 2 ∇u m 2 +a u m α+2 L α+2 ≤ 1 2ν f 2 X ′ (4.2.12)
Next, we introduce the following function

(F (ξ), ξ) := ν(∇ξ, ∇ξ) + a(| ξ | α ξ, ξ) -(f, ξ)
We have

(F (ξ), ξ) ≥ ν ξ 2 +a ξ α+2 L α+2 -f X ′ ξ = (ν ξ -f X ′ ) ξ +a ξ α+2 L α+2 when ξ = K, K > 1 ν f X ′ . The lemma 4.2.
3 gives the existence of u m solution of (4.2.9) for each m. Moreover, the sequence (u m ) is bounded by (4.2.12), then there exists a subsequence denoted (u m ) which

u m ⇀ u in X ∩ L α+2 (Ω) (4.2.13)
From the Sobolev embedding theorem we get

u m -→ u in L 6-ε (4.2.14)
with ε is some positive constant. For the convergence of three quantities in (4.2.9), we proceed as follow; as m -→ ∞ we have

(∇u m , ∇w k ) -→ (∇u, ∇w k ) (4.2.15)
For the two remaining terms, we develop

| ((u m .∇)u m , w k ) -((u.∇)u, w k ) | ≤| ((u m -u).∇)u m , w k ) | + | ((u.∇)(u m -u), w k ) | = T (1) + T (2)
The first quantity can be raised by

T (1) ≤ w k L 6 ∇u m u m -u L 3 ≤ c w k 1 ∇u m u m -u L 3
By (4.2.14) we obtain

lim m→∞ T (1) = 0
It remains to raise the term T (2)

T (2) ≤| ((u.∇)w k , (u m -u)) | ≤ u L 6 ∇w k u m -u L 3 ≤ C w k u m -u L 3
Therefore,

lim m→∞ T (2) = 0
Finally, we obtain the ensuing convergence result

| ((u m .∇)u m , w k ) -((u.∇)u , w k ) |→ 0, m → ∞ (4.2.16)
For the last term, we use

| (| u m | α u m , w k ) -(| u | α u, w k ) | ≤ |(|u m | α (u m -u), w k )| + |((|u m | α -|u| α )u, w k = T (3) + T (4)
Fort the first term T (3) , thank to the embedding theorem

T (3) ≤ Csup|w k |.||u m || α L α+2 ||u m -u|| L 6-ε and according to α = 10 -2ε we obtain lim m→∞ T (3) = 0
The last term T (3) can be raised with the same approach

T (4) ≤ C||u m -u|| L 6-ε .||u m || α L α+2 ||w k || L p For p = ∞ and α = 10 -2ε, we have lim m→∞ T (4) = 0 Therefore lim m→∞ (| u m | α u m , w k ) = (| u | α u, w k ) (4.2.17)
From (4.2.16)-(4.2.17), the velocity u ∈ X satisfies the equation

ν(∇u, ∇w k ) + ((u.∇)u, w k ) + (a | u | α u, w k ) = (f, w k ) (4.2.18)
Finally, we write any ϕ ∈ X as a limit of a linear combination of w k . Then the velocity

u ∈ X satisfies ν(∇u, ∇ϕ) + ((u.∇)u, ϕ) + (a | u | α u, ϕ) = (f, ϕ) f or all ϕ ∈ X (4.2.19)
Using (4.2.12) one gets

ν 2 ∇u +a u α+2 L α+2 ≤ 1 2ν f 2 X ′ ( 4
|||u| α u -|v| α v|| ≤ C||∇(u -v)|| (4.2.24)
After the study of the existence of the solution for the boundary problem, we proceed to treat the uniqueness of the solution.

Uniqueness of weak solution

This theorem states the uniqueness of the weak solution for the boundary problem (4.2.2). Theorem 4.2.2. Let Ω be a bounded domain in R d (d=2, 3), locally Lipschitz, and f ∈ X ′ .

If ν is "big enough" and 2 3 ≤ α ≤ 2, then the weak solution of (4.2.2) is unique.

Proof. Let us suppose that the problem (4.2.2) has two weak solutions u 1 and u 2 . Define ϕ = u 1u 2 and write

ν ∇(u 1 -u 2 ) 2 +(((u 1 -u 2 ).∇)u 1 , u 1 -u 2 ) + (a|u 1 | α u 1 -a|u 2 | α u 2 , u 1 -u 2 ) = 0 which leads to ν ∇(u 1 -u 2 ) 2 +(((u 1 -u 2 ).∇)u 1 , u 1 -u 2 ) +(a|u 1 | α (u 1 -u 2 ), (u 1 -u 2 )) + a((|u 1 | α -|u 2 | α )u 2 , u 1 -u 2 ) = 0 This gives ν ∇(u 1 -u 2 ) 2 = -(((u 1 -u 2 ).∇)u 1 , u 1 -u 2 ) -(a|u 1 | α (u 1 -u 2 ), (u 1 -u 2 )) -a((|u 1 | α -|u 2 | α )u 2 , u 1 -u 2 ) (4.2.25)
The first term on the right hand side in (4.2.25) implies

| -(((u 1 -u 2 ).∇)u 1 , u 1 -u 2 )| ≤ ∇u 1 . u 1 -u 2 2 L 4 ≤ c 1 ν f X ′ u 1 -u 2 2 L 6 (4.2.26)
Using the embedding of

L 6 ֒→ H 1 | -(((u 1 -u 2 ).∇)u 1 , u 1 -u 2 )| ≤ c 1 ν f X ′ u 1 -u 2 2 1 (4.2.27)
We now turn to the other terms using

1 ≤ 3α 2 ≤ α + 2, i.e., 2 3 ≤ α ≤ 4, we obtain | -(a|u 1 | α (u 1 -u 2 ), (u 1 -u 2 ))| = |a|u 1 | α |u 1 -u 2 | 2 | ≤ a |u 1 | α 3 2 |u 1 -u 2 | 2 L 3 ≤ a |u 1 | α 3α 2 u 1 -u 2 2 L 6 ≤ ca u 1 α L α+2 ∇(u 1 -u 2 ) 2 ≤ c 2 a f X ′ ∇(u 1 -u 2 ) 2 (4.2.28)
The last term on the right of (4.2.25), and according to 1

≤ 3α 2 ≤ α + 2, i.e., 2 3 ≤ α ≤ 4 we obtain | -a((|u 1 | α -|u 2 | α )u 2 , (u 1 -u 2 ))| ≤ c 3 aα f X ′ ∇(u 1 -u 2 ) 2 (
ν ∇(u 1 -u 2 ) 2 -( c 1 ν f X ′ +c 2 a f X ′ +c 3 aα f X ′ ) ∇(u 1 -u 2 ) 2 ≤ 0
We conclude, for ν big enough, that u 1 = u 2 .

The established inequalities prove that the velocity u is bounded. In the next subsection we treat the penalized problem related to the Navier Stokes equations with damping (4.2.2). We use a penalty method to overcome the difficulties associated with the constraint "∇.u = 0". Among the method adopted to relax the constraint ∇.u = 0 we cite 1. The penalty method. We refer here to J. Shen [START_REF] Shen | On error estimates of the penalty method for unsteady Navier Stokes equations[END_REF], R. Temam [START_REF] Temam | Une méthode d'approximation de la solution des équations de Navier-Stockes[END_REF] 140CHAPTER 4. ANALYSIS AND FINITE ELEMENT APPROXIMATION OF OPTIMAL CONTROL P

∇.u ε + εp ε = 0 in Ω × [0, T ]
2. Artificial compressibility, we mention here the works of Chorin [START_REF] Joel | Numerical solution for the Navier Stokes equations[END_REF] and of R. Temam [START_REF] Temam | Sur l'approximation de la solution des équations de Navier Stokes par la méthode des pas fractionnaires (ii)[END_REF] ∇.

u ε + εp ε t = 0 in Ω × [0, T ]
3. The projection method

∇.u ε -ε∆p ε = 0 in Ω × [0, T ], ∂p ε ∂n = 0 in ∂Ω × [0, T ]
introduced by Temam [START_REF] Temam | Une méthode d'approximation de la solution des équations de Navier-Stockes[END_REF] and Chorin [START_REF] Joel | Numerical solution for the Navier Stokes equations[END_REF][START_REF] Joel | On the convergence of discrete approximations to the Navier Stodes equations[END_REF] and has a high computational cost.

4. The pseudocompressibility method

∇.u ε -ε∆p ε t = 0 in Ω × [0, T ], ∂p ε ∂n = 0 in ∂Ω × [0, T ]

Perturbed equation

Let us now give the penalized formulation associated to the Navier Stokes equations with damping

ν∆u ε + (u ε .∇)u ε + 1 2 (∇.u ε )u ε + a|u ε | α u ε + ∇p ε = f in Ω εp ε + ∇.u ε = 0 in Ω u ε = 0 on ∂Ω (4.2.30)
The penalized weak formulation of the equation (4.2.30) writes for ε > 0

Find (u ε , p ε ) ∈ X × Y such that ν(∇u ε , ∇v) + ((u ε .∇)u ε , v) + 1 2 ((∇.u ε ), u ε .v) + a(|u ε | α u ε , v) + 1 ε (∇.u ε , ∇.v) = (f, v) ε(p ε , q) + (∇.u ε , q) = 0 (4.2.31) for all v ∈ X and q ∈ Y .
Here are some elementary proprieties of the functional defined by T (w, u, v) = ((w.∇)u, v)+ 1 2 ((∇.w), u.v) for (w, u, v) ∈ X 3 . T is trilinear and there exists a positive constant C > 0 such that

T (w, u, v) ≤ C w 1 u 1 v 1 ∀ (w, u, v) ∈ X 3 (4.2.32)
Moreover, T satisfies

((w.∇)v, v) + 1 2 ((∇.w), v.v) = 0 f or all w, v ∈ X (4.2.33)
and

((w.∇)u, v) + 1 2 ((∇.w), u.v) = -((w.∇)v, u) - 1 2 ((∇.w), v.u) f or all (w, u, v) ∈ X 3 (4.2.34)
We begin with an existence result.

Existence of weak solution of the perturbed problem

The following theorem asserts that the solution for the perturbed problem (4.2.30) exists and it is bounded.

Theorem 4.2.3. Let Ω a bounded domain in R d (d = 2, 3), f ∈ X
′ is a given exterior function. Then the perturbed problem (4.2.30) has a weak solution satisfying

ν 2 ∇u ε 2 +a u ε α+2 L α+2 +ε p ε 2 ≤ 1 2ν f 2 X ′ (4.2.35)
Proof. We consider (w k ) an orthonormal basis of X and (r j ) an orthonormal basis of Y. Let X m =< w 1 , ..., w m > and Y m =< r 1 , ..., r m > for some positive integer m. We define the sequence (u ε m ) and (p ε m )

u ε m = m k=1 ξ km w k , p ε m = m k=1 ψ km r j (4.2.36)
the approximate solution of u ε and p ε such that

ν(∇u ε m , ∇w k ) + (u ε m .∇)u ε m , w k ) + 1 2 (∇.u ε m , u ε m .w k ) + a(|u ε m | α u ε m , w k ) -(p ε m , ∇.w k ) = (f, w k ) f or k = 1, ..., m (4.2.37)
ε(p ε m , r j ) + (∇.u ε m , r j ) = 0, f or j = 1, ..., m (4.2.38)
We multiply by ξ km the equation (4.2.37) and by ψ jm the equation (4.2.37) and sum with respect to k and j to get

ν ∇u ε m 2 +a u ε m α+2 L α+2 +ε p ε m 2 = (f, u ε m ) ≤ f X ′ ∇u ε m (4.2.39)
Using the Hölder inequality to have

ν 2 ∇u ε m 2 +a u ε m α+2 L α+2 +ε p ε m 2 ≤ 1 2ν f 2 X ′ (4.2.40) 142CHAPTER 4

. ANALYSIS AND FINITE ELEMENT APPROXIMATION OF OPTIMAL CONTROL P

For the convergence of the all terms on the left hand side in (4.2.37), we follow a similar process as in the previous section.

u ε satisfies ν(∇u ε , ∇w k ) + (u ε .∇)u ε , w k ) + 1 2 (∇.u ε , u ε .w k ) + a(|u ε | α u ε , w k ) -(p ε , ∇.w k ) = (f, w k ) f or k = 1, ..., m (4.2.41)
For all ϕ ∈ H 1 (Ω) written as a linear combination of w k , u ε and p ε satisfy

ν(∇u ε , ∇ϕ) + (u ε .∇)u ε , ϕ) + 1 2 (∇.u ε , u ε .ϕ) + a(|u ε | α u ε , ϕ) -(p ε , ∇.ϕ) = (f, ϕ) ∀ ϕ ∈ H 1 (Ω) (4.2.42)
Using (4.2.40) to infer that

ν 2 ∇u ε 2 +a u ε α+2 L α+2 +ε p ε 2 ≤ 1 2ν f 2 X ′ (4.2.43)
In the following section, we firstly use the fact that u ε is bounded to show that p ε is bounded. Next, We extract a subsequence convergent to the boundary problem solution. Finally, we establish the error estimates.

Error estimates

We consider that (u ε , p ε ) are solutions of the problem (4.2.30) and let s be the norm

s = sup u,v,w∈X | ((w.∇)u.v) + 1 2 (∇.w, u.v) | w 1 u 1 v 1 (4.2.44)
Then, the data ν and f are assumed to satisfy

s ν 2 f X ′ < 1 (4.2.45)
The equation (4.2.45) can be written differently; there is a fixed r with 0

< r < 1 such that s ν 2 f X ′ ≤ 1 -r (4.2.46)
This lemma gives the convergence of non linear terms which will be used in this section.

Lemma 4.2.5. If u k → u ∈ X. Then for all v ∈ X lim k→∞ ((u k .∇)u k , v) = ((u.∇)u, v) lim k→∞ (∇.u k , u k .v) = (∇.u, u.v) lim k→∞ (∇.u k , ∇.v) = (∇.u, ∇.v) lim k→∞ (|u k | α |u k , v) = (|u| α |u, v) (4.2.47)
Proof. For the last term, we use Lemma 4.2.4

(|u k | α u k -|u| α u) ≤ ∇(u k -u)
We introduce the inf-sup condition: there exists a constant β > 0 such that

sup v∈X(v =0) (q, ∇.v) v 1 ≥ β q Z f or all q ∈ Y (4.2.48)
where

q = {q ∈ Y |q -q = constant} , Z = {q|q ∈ Y } and q Z = inf x∈R q + x (4.2.49)
We also define the operators

B ∈ L(X, Y ′ ) and B * ∈ L(Y, X ′ ) by (q, ∇.v) = (q, Bv) Y ×Y ′ = (B * q, v) X ′ ×X (4.2.50)
We have

Bv = ∇.v and B * q = ∇q (4.2.51)
and

KerB * = {q ∈ Y |(q, ∇.v) = 0 f or all v ∈ X} = {constants}
The ensuing theorem shows that the perturbed pressure is bounded. The convergence of the couple (u ε , p ε ) is asserted by the extraction of a subsequence convergent to the solution for the boundary problem (4.2.2). We also give a priori estimate for the velocity and the pressure.

Theorem 4.2.4. Let (u ε , p ε ) be solution of the problem (4.2.30) and (u,p) the solution in X × Y /R of the problem (4.2.2). Then we have

(p ε ) ∈ Y is unif ormly bounded in ε (4.2.52)
and as ε -→ 0. u ε → u in X (4.2.53)

and p ε → p in Y (4.2.54)
where p is the orthogonal projection in Y of p onto (KerB * ) ⊥ . And we have the following estimates

u ε -u 1 + p ε -p ≤ Cε (4.2.55)
with C is a positive constant non depending of ε.

-(p ε , ∇.v) = (f, v) -ν(∇u ε , ∇v) -((u ε .∇)u ε , v) - 1 2 (∇u ε , u ε .v) -(a|u ε | α u ε , v) for all v ∈ X
Thanks to (4.2.32) and Lemma 4.2.4

|(p ε , ∇.v)| ≤ ( f X ′ +ν u ε 1 +C 1 u ε 2 1 +C 2 a u ε 1 )
v 1 for all v ∈ X Applying this in the inf-sup condition with q = p ε ∈ Y , and using the fact that ∇u ε is uniformly bounded, we get

p ε Z ≤ C β (4.2.56)
with C is a constant no depending of ε. Using the second equation of (4.2.30) and the definition of kerB * (p ε , q) = 0 for all q ∈ KerB * Then

p ε ∈ (KerB * ) ⊥ in Y Since p ε Z = p ε Therefore p ε ≤ C β (4.2.57) (p ε ) is uniformly bounded in ε.
The couple (u ε , p ε ) is uniformly bounded in X × Y . Then there exists a subsequence without lose of generality denoted (u ε , p ε ) convergent to (u, p) ∈ X × Y . We know already that (kerB * ) ⊥ is closed and convex that is gives (kerB * ) ⊥ is weakly closed and p ε ∈

(kerB * ) ⊥ → p ∈ (kerB * ) ⊥ .
Let us now show that (u, p) is solution of the problem (4.2.2). We use the second equation of (4.2.30), we have

(q, ∇.u ε ) = -ε q, - 1 ε ∇.u ε = -ε(q, p ε ) Therefore |(q, ∇.u ε )| ≤ ε q p ε (4.2.58)
And by lemma (4.2.5), ∇.u ε → ∇.u in L 2 (Ω). we get (q, ∇.u ε ) → (q, ∇.u) as ε → 0

Passing to limits in (4.2.58) and using (4.2.57) yields (q, ∇.u) = 0 for all q ∈ Y Therefore the second equation of (4.2.2) is verified. Using lemma 4.2.5 and the previous result gives

lim ε→0 (∇u ε , ∇v) = (∇u, ∇v) f or all v ∈ X lim ε→0 ((u ε .∇)u ε , ∇v) = ((u.∇)u, ∇v) f or all v ∈ X lim ε→0 (∇.u ε , u ε .v) = 0 f or all v ∈ X lim ε→0 (| u ε | α u ε , ∇v) = (| u | α u, ∇v) f or all v ∈ X lim ε→0 (p ε , ∇.v) = (p, ∇.v) f or all v ∈ X (4.2.59)
We replace these limits in the first equation (4.2.30) implies that the limit of (u ε , p ε ) is solution of the problem (4.2.2). Finally, it remains to establish the estimate (4.2.55). Subtracting the first equation (4.2.2) from the first equation of (4.2.30) and using the second equation of (4.2.30) gives

(p ε -p, ∇.v) = ν(∇(u ε -u), ∇v) +((u ε .∇)u ε , v) -((u.∇)u, v) + 1 2 (∇.u ε , u ε .v) +(a | u ε | α u ε , v) -(a | u | α u, v) ∀v ∈ X (4.2.60) Taking v = u ε -u yields (p ε -p, ∇.(u ε -u)) = ν u ε -u 2 1 +((u ε .∇)u ε , u ε -u) + 1 2 (∇.u ε , u ε .(u ε -u)) -((u.∇)u, (u ε -u)) + a(|u ε | α u ε , (u ε -u)) -a(|u| α u, (u ε -u)) (4.2.61
) Rewriting the fourth term on the right hand side in (4.2.61),

-((u.∇)u, u ε -u) = (((u ε -u -u ε ).∇)(u ε -u + u ε ), (u ε -u)) = (((u ε -u).∇)u ε , (u ε -u)) + (((u ε -u).∇)(u -u ε ), (u ε -u)) -((u ε .∇)u ε , (u ε -u)) -((u ε .∇)(u -u ε ), (u ε -u)) We note that -((u.∇)u, u ε -u) = (((u ε -u).∇)u ε , (u ε -u)) -((u ε .∇)u ε , (u ε -u)) 1 2 (∇.(u ε -u), u ε .(u ε -u)) - 1 2 (∇.u ε , u ε .(u ε -u)) (4
(p ε -p, ∇.(u ε -u)) = ν u ε -u 2 1 +(((u ε -u).∇)u ε , (u ε -u)) + 1 2 (∇.(u ε -u), u ε .(u ε -u)) + a(|u ε | α u ε -|u| α u, (u ε -u)) ≥ ν u ε -u 2 1 -s u ε -u 2 1 u ε 1 (4.2.63)
We use the fact that (u ε ) is bounded and by the inequality (4.2.46) we obtain

u ε 1 ≤ ν s (1 -r)
Consequently

((p ε -p), ∇.(u ε -u)) ≥ νr u ε -u 2 1
By the second equation in (4.2.30) we have

u ε -u 2 1 ≤ ε νr (p -p ε , p ε ) = ε νr (p -p ε , p) - ε νr p -p ε 2 (4.2.64) Therefore u ε -u 2 1 ≤ ε νr p -p ε p . (4.2.65)
We now observe that

((u ε .∇)u ε , v) = (((u ε .∇)u ε , v) + ((u.∇)u ε , v) (4.2.66) -((u.∇)u, v) = ((u.∇)(u ε -u), v) -((u.∇)u ε , v) (4.2.67) 1 2 ∇.u ε , u ε .v) = 1 2 (∇.(u ε -u), u ε .v) (4.2
(p ε -p, ∇.v) = ν(∇(u ε -u), ∇v) + (((u ε -u).∇)u ε , v) + ((u.∇)(u ε -u), v) 1 2 (∇.(u ε -u), u ε .v) + a(|u ε | α u ε -|u| α u, v) (4.2.69)
Using (4.2.69) in the condition inf-sup (4.2.48) gives with q = pp

p ε -p Z ≤ 1 β (ν u ε -u 1 +s( u ε 1 + u 1 ) u ε -u 1 +C 1 a u ε -u 1 )
Note that p ε , p ∈ (kerB * ) ⊥ , and from (4.2.46) we obtain

p ε -p ≤ 1 β (ν + 2(1 -r)ν + C 1 a) u ε -u 1 Therefore p ε -p ≤ A β u ε -u 1 (4.2.70)
where A = (3 -2r)ν + C 1 a. Putting (4.2.70) in (4.2.65) to have

u ε -u 1 ≤ A νrβ p ε. (4.2.71)
Finally, we substitute (4.2.71) in (4.2.70) to infer that

p ε -p ≤ A 2 νrβ 2 p ε. (4.2.72)
After the study of different proprieties about the boundary and penalized problem, we pass to the shape optimization. We introduce in the ensuig section the shape optimization problem. We derive an adjoint equation related to the perturbed equation. Finally, we compute the gradient of the cost function in terms of state and adjoint variables.

Control problem

During this section, we introduce a shape optimization problem to minimize an objective function. We derive an adjoint system related to the coming penalized problem

-ν∆u ε + (u ε .∇)u ε + 1 2 (∇.u ε )u ε + a | u ε | α u ε + ∇p ε = f in Ω ∇.u ε + εp ε = 0 in Ω u ε = 0 on Γ 0 u ε = g on Γ 1 ν ∂u ε ∂n -p ε n = 0 on Γ 2 (4.3.1)
and compute the shape gradient of the cost function in terms of state and adjoint variables.

For simplicity we assume that f = 0. We first define the objective function

J(Ω) = J 1 (Ω) + J 2 (Ω) (4.3.2)
where We introduce also the following functions

J 1 (Ω) = 1 2 Ω |u ε -u d | 2 dx, and J 2 (Ω) = σ 2 Ω |curl(u ε )| 2 dx
F (Ω, u, p, v, q) = Ω ν∇u ε : ∇v + (u ε .∇)u ε .v + 1 2 (∇.u ε )u ε .v + a|u ε | α u ε .v -p ε ∇.v dx - Ω q∇.u ε dx -ε Ω qp ε dx
Then the optimization problem can be expressed as

min Ω∈Ω ad J(Ω) = 1 2 Ω |u ε -u d | 2 dx + σ 2 Ω |curl(u ε )| 2 dx
such that (u ε , p ε ) satisf ies the euqtaion (4.3.1) (4.3.3)

Adjoint system

We begin by deriving an adjoint system associated with the penalized system (4.3.1). This theorem describes the adjoint system related to the general penalized problem (4.3.1).

Theorem 4.3.1. Let (u ε , p ε ) ∈ Vg (Ω) × Y . The adjoint system associated to the equation (4.3.1) takes the form

-ν∆v + (∇u ε ) T .v -(u ε .∇)v - 1 2 ∇(u ε .v) + 1 2 (∇.u ε )v + a|u ε | α-2 (|u ε | 2 v + α(u ε .v)u ε ) +∇q = (u ε -u d ) -σ curl(curl(u ε )) on Ω ∇.v + εq = 0 on Ω v = 0 on Γ 0 ∪ Γ 1 ν ∂v ∂n + 2(u ε .n)v -nq = σcurl(u ε ).τ on Γ 2 (4.3.4)
Proof. Let L(Ω, u ε , p ε , v, q) be a Lagrangian functional defined by

L(Ω, u ε , p ε , v, q) = J(Ω) -F (Ω, u ε , p ε , v, q) (4.3.5)
We first derive L with respect to the state variable p in any direction p ∈ M to obtain

∂L ∂p ε (Ω, u ε , p ε , v, q).p = Ω p∇.v dx + Ω εpq dx
The variation p is arbitrary, we get

∇.v + εq = 0 on Ω (4.3.6)
Next, we derive L with respect of the state variable u ε in the arbitrary direction ũ ∈ V 0 (Ω) to have

0 = ∂L ∂u ε (Ω, u ε , p ε , v, q).ũ = Ω (u ε -u d ).ũdx + σ Ω curl(u ε ).curl(ũ)dx - Ω ν∇ũ : ∇v + (ũ.∇)u ε .v + (u ε .∇)ũ.v + 1 2 (∇.ũ)u ε .v + (∇.u ε )ũ.v dx + Ω q∇.ũ dx - Ω a|u ε | α ũ + aα|u ε | α-2 (u ε .ũ)u ε .v dx = Ω [(u ε -u d ) -σ curl(curl(u ε )) + ν∆u ε -(∇u ε ) T .v + (u ε .∇)v + 1 2 ∇(u ε .v) - 1 2 (∇.u ε )v -(a|u ε | α v + aα|u ε | α-2 (u ε .v)u ε -∇q].ũ dx + Γ 2 σcurl(u ε )τ -ν ∂v ∂n + 2(u ε .n)v -nq .ũds
Considering an arbitrary direction ũ which vanishes in neighbourhood of the boundary Γ 2 , we write

-ν∆v + (∇u ε ) T .v -(u ε .∇)v - 1 2 ∇(u ε .v) + 1 2 (∇.u ε )v + a|u ε | α-2 (|u ε | 2 v + α(u ε .v)u ε ) +∇q = (u ε -u d ) -σ curl(curl(u ε ))
An arbitrary ũ in Γ 2 gives

ν ∂v ∂n + 2(u ε .n)v -nq = σcurl(u ε ).τ on Γ 2
Finally we obtain the adjoint system

-ν∆v + (∇u ε ) T .v -(u ε .∇)v - 1 2 ∇(u ε .v) + 1 2 (∇.u ε )v + a|u ε | α-2 (|u ε | 2 v + α(u ε .v)u ε ) +∇q = (u ε -u d ) -σ curl(curl(u ε )) on Ω ∇.v + εq = 0 on Ω v = 0 on Γ 0 ∪ Γ 1 ν ∂v ∂n + 2(u ε .n)v -nq = σcurl(u ε ).τ on Γ 2 (4.3.7) j(t) = min (u ε ,p ε )∈Vg(Ω)× Ŷ (Ω) max (v,q)∈V 0 (Ω)× Ŷ (Ω) L(Ω t , u ε • T -1 t , p ε • T -1 t , v • T -1 t , q • T -1 t )
We introduce the following functions which depend on the parameter t

l 1 (t) = 1 2 Ωt |u ε • T -1 t -u d | 2 dx + σ 2 Ω |curl(u ε • T -1 t )| 2 dx, l 2 (t) = Ωt [ν∇(u ε • T -1 t ) : ∇(v • T -1 t ) + ((u ε • T -1 t ).∇)(u ε • T -1 t ).(v • T -1 t ) + a|u ε • T -1 t | α (u ε • T -1 t ).(v • T -1 t ) -(p ε • T -1 t )∇.(v • T -1 t )]dx - Ωt (q • T -1 t )∇.(u ε • T -1 t )dx -ε Ωt (q • T -1 t )(p ε • T -1 t )dx
The Lagrangian functional writes

L(Ω t , u ε • T -1 t , p ε • T -1 t , v • T -1 t , q • T -1 t ) = l 1 (t) -l 2 (t) If Φ : [0, τ ] × R 2 →
R is sufficiently smooth, we have the following Hadamard formula

d dt Ωt Φ(t, x)dx| t=0 = Ω ∂Φ ∂t (0, x)dx + ∂Ω Φ(0, x)V (0, X).nds (4.3.9)
Let V(0, X) ∈ V ad , and observe that V(0, X) = V. Therefore we can derive the shape gradient using the formula (4.3.9):

d dt L(Ω t , u ε • T -1 t , p ε • T -1 t , v • T -1 t , q • T -1 t )| t=0 = l ′ 1 (0) -l ′ 2 (0) (4.3.10)
where

l ′ 1 (0) = Ω (u ε -u d ).(-∇u ε .V)dx + σ Ω curl(u ε )curl(-∇u ε .V)dx + 1 2 Γ 0 (|u ε -u d | 2 V.n)ds + σ 2 Γ 0 (|curl(u ε )| 2 V.n)ds (4.3.11) l ′ 2 (0) = Ω [ν∇(-∇u ε .V) : ∇v + ν∇u ε : ∇(-∇v.V) + ((-∇u ε .V).∇u ε ).v + (u ε .∇(-∇u ε .V).v + (u ε .∇u ε ).(-∇v.V) 1 2 ∇.(-∇u ε .V)u ε .v + 1 2 ∇.u ε (-∇u ε .V).v + 1 2 ∇.u ε u ε .(-∇v.V) + aα|u ε | α-2 (-∇u ε .V.u ε )(u ε .v) + a|u ε | α (-∇u ε .V).v + a|u ε | α u ε .(-∇v.V) -(∇p.V)∇.v -p∇.(-∇v.V) -q∇.(-∇u ε .V) -(∇.u ε )(-∇q.V)]dx
152CHAPTER 4. ANALYSIS AND FINITE ELEMENT APPROXIMATION OF OPTIMAL CONTROL P Using Green's formula and the condition u ε = 0 on Γ 0 , we obtain

l ′ 2 (0) = - Ω [(-ν∆u ε + (u ε .∇)u ε + 1 2 ∇.u ε u ε + a|u ε | α u ε + ∇p).(∇v.V)]dx + Ω (∇.u ε )(∇q.V)dx + Ω (∇.v)(∇p.V)dx + Ω [-ν∆v + (∇u ε ) T .v -(u ε .∇)v - 1 2 ∇(u ε .v) + 1 2 (∇.u ε )v + a|u ε | α-2 (|u ε | 2 v + α(u ε .v)u ε ) -∇q].(∇u ε .V)dx - Γ 0 [ν ∂v ∂n + 2(u ε .n)v -nq].(∇u ε .V)ds - Γ 0 [ ∂u ε ∂n -nq].(∇v.V)ds + Γ 0 [ν∇u ε : ∇v + (u ε .∇u ε ).v + a|u ε | α u ε .v -p∇.v -q∇.u ε ]V.n ds (4.
3.12) Replacing (4.3.11) and (4.3.12) in (4.3.10) and using the fact that (u ε , p ε ) is solution of (4.3.1) and (v,q) is solution of (4.3.4) respectively yields to

dJ(Ω; V) = d dt L| t=0 = 1 2 Γ 0 ((|u ε -u d | 2 + σ|curl(u ε )| 2 )V.n)ds + Γ 0 [ν ∂v ∂n + nq].(∇u ε .V)ds + Γ 0 [ν ∂u ε ∂n -np].(∇v.V)ds - Γ 0 [(ν∇u ε : ∇v)V.n ds -σ Γ 0 ( ∂u ε ∂n .(curl(u ε ) ∧ n))V.n ds (4.3.13) Note that u ε = 0, v=0 in Γ 0 we have n.(∇u ε .V) = ∇u ε .(n ⊗ n).V.n = ∇u ε .n.n(V.n) = (∇.u ε )(V.n) = 0, ∀x ∈ Γ 0 (4.3.14) ∂v ∂n .(∇u ε .V) = ∇u ε .(n ⊗ n).V. ∂v ∂n = ∂u ε ∂n . ∂v ∂n (V.n) = (∇u ε : ∇v)V.n (
dJ(Ω; V) = Γ 0 1 2 |u ε -u d | 2 + σ 2 |curl(u ε )| 2 + ∂u ε ∂n . 2ν ∂v ∂n -σcurl(u ε ) ∧ n V.n ds (4.
3.17) Consequently, the shape gradient writes

∇J = 1 2 |u ε -u d | 2 + σ 2 |curl(u ε )| 2 + ∂u ε ∂n . 2ν ∂v ∂n -σcurl(u ε ) ∧ n n (4.3.18)
After the computation of the gradient of the objective function and the derivation of the adjoint system, we adopt a discrete finite element method to discretize the penalized problem. We establish an error estimates to prove the convergence of the discrete approximate solution to the boundary problem solution.

Finite element method

We consider a finite element space Ω h of Ω. Let Ωh = ∪ k Tk , with {T k } is a partition of Ω into non overlapping elements. Let us introduce the finite element dimensional space X h of X = (H 1 0 (Ω)) d , d=2 or 3 as follow

X h = v ∈ (C( Ω)) d | v |T k ∈ Q k f or all T k ⊂ Ω h (4.4.1)
where Q k is the space of polynomials functions of degree k given by

Q k = span d i=1 x b i i | 0 ≤ b i ≤ k (4.4.2)
We define the integration rule I(.) by

Ω gdx ≃ I(g) = E k=1 G k i=1 g(x k i )w k i (4.4.3)
where x k i ∈ Ωk , are the integration point coordinates,

w k i , 1 ≤ i ≤ G k , 1 ≤ k ≤ E,
are the integration weights of the Gauss rule I(.) on any element Ω k and E is the number of elements in the mesh. The pressure space Y h is piecewise discontinuous defined with the points in the integration above. We have for any q h ∈ Y h

I(q h ∇.v h ) = (q h , ∇.v h ) f or all v h ∈ X h (4.4.4)
For f ∈ (X h ) ′ , and for ε > 0 fixed we introduce the approximate perturbed problem reads as Find u ε h ∈ X h such that

ν(∇u ε h , ∇v) + ((u ε h .∇)u ε h , v h ) + 1 2 (∇.u ε h , u ε h .v h ) + a(|u ε h | α u ε h , v h ) + 1 ε I((∇.u ε h )(∇.v h )) = (f, v h ) f or all v h ∈ X h (4.4.5)
Moreover, the approximate pressure verifies

p ε h (x e i ) = - 1 ε ∇.u ε h (x e i ), 1 ≤ i ≤ G, 1 ≤ e ≤ E ( 4
h ν f (X h ) ′ ≤ s ν 2 f X ′ ≤ 1 -r (4.4.7)
The next result focuses on the existence of a weak solution for the approximate perturbed problem (4.4.5).

Theorem 4.4.1. Assume that the regularity of the data (4.4.7) holds for h, ε > 0. We have the following proprieties 1. There exists a solution to the approximate perturbed problem (4.4.5).

2. The sequence (u ε h ) ∈ X h is uniformly bounded in ε.

The following estimates

ν 2 ∇u ε h 2 +a u ε h α+2 L α+2 +ε p ε h 2 ≤ 1 2ν f 2 X ′ (4.4.8)
hold.

Proof. We follow similar steps as for the continuous problem in theorem 4.2.1 to obtain

u ε h X h ≤ 1 ν f (X h ) ′ (4.4.9)
We set the inf-sup stability condition for the approximate problem. We suppose, there exists a constant β h > 0, such that

sup v h ∈X h I(q h ∇.v h ) v h V ≥ β h qh Z h f or all q h ∈ Y h (4.4.10)
We introduce the finite element space Z h with

qh = qh ∈ Y h |q h -q h ∈ KerB * h (4.4.11)
We also define the operators

B h ∈ L(X h , (Y h ) ′ ), B * h ∈ L(Y h , (X h ) ′ ) by I(q h ∇.v h ) = (q h , B h v h ) (Y h ) ′ ×Y h = (B * h q h , v h ) (X h ) ′ ×X h , f or all (v h , q h ) ∈ X h × Y h (4.4.
12) The kernel of the operator B * h is defined by

KerB * h = q h ∈ Y h | I(q h ∇.v h ) = 0 ∀v h ∈ X h (4.4.13)
The norm on Z h in (4.4.10) writes

qh Z h = inf q 0 h ∈KerB * h q 0 h + q h (4.4.14)
The main idea of these results is that the pressure is bounded in ε and that the solution for the approximate penalized problem (4.4.5) converges to the solution of a boundary approximate problem. (ii) The couple (u ε h , p ε h ) converges to the solution (u h , p h ) of the problem

Find (u h , p h ) ∈ X h × Y h such that ν(∇u h , ∇v h ) + ((u h .∇)u h , v h ) + 1 2 ((∇.u h ), u h .v h ) + a(|u h | α u h , v h ) -(p h , ∇.v h ) = (f, v h ) f or all v h ∈ X h (q h , ∇.u h ) = 0 f or all q h ∈ Y h (4.4.15)
Proof. The theorem 4.4.1 ensures that (u ε h ) ∈ X h is uniformly bounded in ε. Replacing (4.4.6) in (4.4.5)

I(p ε h ∇.v h ) = -(f, v h ) + ν(∇u ε h , ∇v) + ((u ε h .∇)u ε h , v h ) + 1 2 (∇.u ε h , u ε h .v h ) + a(|u ε h | α u ε h , v h
) Substituting this results in the discrete inf-sup condition (4.4.10), Since C is independent of ε.

pε h Z h ≤ 1 β h f (X h ) ′ +ν u ε h 1 +c 1 u ε h 2 1 +c 2 u ε h 1 ≤ C β h . ( 4
The sequence (u ε h , p ε h ) is bounded, then there exists a subsequence denoted (u ε h , p ε h ) without lose of generality which converge to (u h , p h ) ∈ X h × Y h . And we have

lim ε→0 I(q h ∇.u ε h ) = I(q h ∇u h ), f or all q h ∈ Y h (4.4.18)
Using (4.4.6),

I(q h ∇.u ε h ) = -εI(q h p ε h ) ≤ ε q h p ε h
We use the fact that (p ε h ) is bounded, and we pass to limits as ε → 0 to obtain

I(q h ∇.v h ) = 0
lim ε→0 ν(∇u ε h , ∇v h ) = ν(∇u h , ∇v h ) f or all v h ∈ X h lim ε→0 ((u ε h .∇)u ε h , v h ) = ((u h .∇)u h , v h ) f or all v h ∈ X h lim ε→0 1 2 (∇.u ε h , u ε h .v h ) = 1 2 (∇.u h , u h .v h ) f or all v h ∈ X h lim ε→0 (|u ε h | α u ε h , v h ) = (|u h | α u h , v h ) f or all v h ∈ X h lim ε→0 (p ε h , ∇.v h ) = (p h , ∇.v h ) f or all v h ∈ X h
We are now in a position to give an priori estimates of the error for the solution of the discrete penalized problem (4.4.5) and the solution of the approximate state problem (4.4.15).

Lemma 4.4.1. Consider (u ε h , p ε h ) ∈ X h × Y h defined by (4.4.5) and (u h , p h ) ∈ X h × Y h solution of the problem (4.4.15). Then u ε h -u h 1 ≤ ε C 1 β h p h (4.4.19)
and

p ε h -p h ≤ ε C 2 β 2 h p h (4.4.20)
where C 1 and C 2 are two positive constant non depending in ε.

Proof. We use the definition of the discrete pressure in (4.4.6) in the equation (4.4.5) and we subtract (4.4.15) to have

ν(∇(u ε h -u h ), ∇v h ) + ((u ε h .∇)u ε h , v h ) + 1 2 (∇.u ε h , u ε h .v h ) + a(|u ε h | α u ε h -|u h | α u h , v h ) -((u h .∇)u h , v h ) - 1 2 (∇.u h , u h .v h ) + (p h -p ε h , ∇.v h ) = 0 f or all v h ∈ X h (4.4.21) With some calculus, for v h ∈ X h we obtain ((u ε h .∇)u ε h , v h ) = (((u ε h -u h ).∇)u ε h , v h ) + ((u h .∇)u ε h , v h ), (4.4.22) -((u h .∇)u h , v h ) = -((u h .∇)(u h -u ε h ), v h ) -((u h .∇)u ε h , v h ) (4.4.23) 1 2 ((∇.u ε h ), u ε h .v h ) = 1 2 (∇.(u ε h -u h ), u ε h .v h ) + 1 2 (∇.u ε h , u ε h .v h ), (4.4.24)
- 1 2 ((∇.u h ), u h .v h ) = 1 2 ((∇.u h ), (u ε h -u h ).v h ) - 1 2 (∇.u h , u ε h .v h ) (4
(p ε h -p h , ∇.v h ) = ν(∇(u ε h -u h ), ∇v h ) + (((u ε h -u h ).∇)u ε h , v h ) + ((u h .∇)(u ε h -u h ), v h ) 1 2 (∇.(u ε h -u h ), u ε h .v h ) + 1 2 (∇.u h , (u ε h -u h ).v h ) + a(|u ε h | α u ε h -|u| α u h , v h ) (4.
pε h -ph Z h ≤ ν + 2ν(1 -r) + aC 3 β h u ε h -u h 1 ( 4
p ε h -p h ≤ ν(3 -2r) + aC 3 β h u ε h -u h 1 (4.4.28) Consider v h = u ε h -u h in the equation (4.4.21) to write (p ε h -p h , ∇.(u ε h -u h )) = ν u ε h -u h 2 1 +((u ε h .∇)u ε h , u ε h -u h ) + 1 2 (∇.u ε h , u ε h .(u ε h -u h ))) -((u h .∇)u h , u ε h -u h )) - 1 2 ((∇.u h , u h .(u ε h -u h )) +a(|u ε h | α u ε h , u ε h -u h ) -a(|u h | α u h , u ε h -u h ) (4.4.29) Using u h = -(u ε h -u h -u ε h
) and thanks to (5.2.13), and the inequality (4.4.27), (4.4.29) takes the form

(p ε h -p h , ∇.(u ε h -u h )) = ν u ε h -u h 2 1 +(((u ε h -u h ).∇)u ε h , (u ε h -u h )) + 1 2 ((∇.(u ε h -u h ), u ε h .(u ε h -u)) +a(|u ε h | α u ε h , u ε h -u h ) -a(|u h | α u h , u ε h -u h ) ≥ ν u ε h -u h 2 1 -s u ε h -u h 1 u ε h 1
. By (4.4.7) and (4.4.9), we infer

(p ε h -p h , ∇.(u ε h -u h )) ≥ νr u ε h -u h 2 1
Using (4.4.15)

u ε h -u h 2 1 ≤ 1 νr (p ε h -p h , ∇.u ε h ) 158CHAPTER 4.

ANALYSIS AND FINITE ELEMENT APPROXIMATION OF OPTIMAL CONTROL P

The use of the property (4.4.6) gives

u ε h -u h 2 1 ≤ ε νr (p ε h -p h , p ε h ) = ε νr (p ε h -p h , p h ) - ε νr p ε h -p h 2 Therefore u ε h -u h 2 1 ≤ ε νr p ε h -p h p h ( 4
u h -u 1 ≤ 3 r -1 + a νr C 1 1 + C β h u -v h 1 + C 2 νr p -q h (4.4.33)
for all v h ∈ X h and all q h ∈ Y h , and

p -ph Z h ≤ ν(3 -2r) + aC β h u -u h 1 + 1 + C ′ β h p -q h f or all q h ∈ Y h (4.4.34)
Proof. We introduce the finite space Subtracting the quantity

X h = v h ∈ X h | (q h , ∇.v h ) = 0 f or all q h ∈ Y h . Since u h ∈ X h and v h ∈ X h , let v h = u h -v h in (4.4.15) to get ν(∇u h , ∇(u h -v h ))+ ((u h .∇)u h , u h -v h ) + 1 2 (∇.u h , u h .(u h -v h )) +a(|u h | α u h , u h -v h ) = (f, u h -v h ) f or all v h ∈ X h
ν(∇v h , ∇(u h -v h )) + ((v h .∇)v h , u h -v h ) + 1 2 (∇.v h , v h .(u h -v h )) + a(|v h | α v h , u h -v h )
from the both sides of (4.4.35), multiplying (4.2.2) by v = u hv h and substituting here in the right hand side

ν(∇(u h -v h ), ∇(u h -v h )) + ((u h .∇)u h , u h -v h ) + 1 2 (∇.u h , u h .(u h -v h )) +a(|u h | α u h , u h -v h ) -((v h .∇)v h , u h -v h ) - 1 2 (∇.v h , v h .(uh -v h )) -a(|v h | α v h , u h -v h ) = ν(∇u, ∇(u h -v h )) + ((u.∇)u, u h -v h ) + a(|u| α u, u h -v h ) -(p, ∇.(u h -v h )) -ν(∇v h , ∇(u h -v h )) -((v h .∇)v h , u h -v h ) - 1 2 (∇.v h , v h .(u h -v h )) -a(|v h | α v h , u h -v h ) f or all v h ∈ X h (4.4.36) Or, v h ∈ X h , (p, ∇.(u h -v h )) = (p-q h , ∇.(u h -v h )).
Adding and subtracting ((u.∇)v h , u hv h ) on the right hand side in the previous result

ν(∇(u h -v h ), ∇(u h -v h )) + ((u h .∇)u h , u h -v h ) + 1 2 (∇.u h , u h .(u h -v h )) +a(|u h | α u h -|v h | α v h , u h -v h ) -((v h .∇)v h , u h -v h ) - 1 2 (∇.v h , v h .(u h -v h )) = ν(∇(u -v h ), ∇(u h -v h )) + ((u.∇)(u -v h ), u h -v h ) + a(|u| α u -|v h | α v h , u h -v h ) -(p -q h , ∇.(u h -v h )) + ((u -v h .∇)v h , u h -v h ) - 1 2 (∇.(u -v h ), v h .(u h -v h )) (4.4.37
) for all v h ∈ X h . The first part of the equality above can be simplified as

ν(∇(u h -v h ), ∇(u h -v h )) + ((u h .∇)u h , u h -v h ) + 1 2 (∇.u h , u h .(u h -v h )) +a(|u h | α u h -|v h | α v h , u h -v h ) -((v h .∇)v h , u h -v h ) - 1 2 (∇.v h , v h .(u h -v h )) = ν(∇(u h -v h ), ∇(u h -v h )) + ((u h .∇)u h , u h -v h ) + 1 2 (∇.u h , u h .(u h -v h )) -(((v h -u h ).∇)(v h -u h ), u h -v h ) -((u h .∇)(v h -u h ), u h -v h ) -(((v h -u h ).∇)u h , u h -v h ) -((u h .∇)u h , u h -v h ) - 1 2 (∇.(v h -u h ), (v h -u h ).(u h -v h ))) - 1 2 ((∇.u h ), (v h -u h ).(u h -v h )) - 1 2 (∇.(v h -u h ), u h .(u h -v h )) - 1 2 (∇.u h , u h .(u h -v h )) +a(|u h | α u h -|v h | α v h , u h -v h ) = ν(∇(u h -v h ), ∇(u h -v h )) + (((u h -v h ).∇)u h , u h -v h ) + 1 2 (∇.(u h -v h ), u h .(u h -v h )) +a(|u h | α u h -|v h | α v h , u h -v h ) (4.4.38) ≥ ν u h -v h 2 1 -s u h -v h 2 1 u h 1 ≥ νr u h -v h 2
u h -v h 1 ≤ 1 r 1 + s ν ( u 1 + v h 1 ) + a ν C 1 u -v h 1 + C 2 νr p -q h (4.4.40)
for all v h ∈ X h and q h ∈ Y h . Simplifying (4.4.40) gives

u -u h 1 ≤ 1 + 1 r 1 + s ν ( u 1 + v h 1 ) + a ν C 1 u -v h 1 + C 2 νr p -q h (4.4.41)
We define v h = h u as the orthogonal projection of u on X h . We have

u -h u 1 = inf v h ∈X h u -v h 1
and h u 1 ≤ u 1 Using these properties in (4.4.41) and the fact that (u) is bounded yields

u -u h 1 ≤ 3 r -1 + a νr C 1 inf v h ∈X h u -v h 1 + C 2 νr p -q h (4.4.42)
It remains to estimate the term inf

v h ∈X h u -v h 1 .
For this end, we solve an auxiliary approximate penalized variational Stokes problem:

let v h ∈ X h , for all ε > 0, Find z ε h ∈ X h such that (∇z ε h , ∇w h ) - 1 ε I((∇.z ε h )(∇.w h )) = (∇v h , ∇w h ) f or all w h ∈ X h (4.

4.43)

There exists an unique solution z ε h ∈ X h of the problem (4.4.43). We define the associated pressure

π ε h ∈ Y h as π ε h (x e i ) = - 1 ε ∇.z ε h (x e i ), 1 ≤ i ≤ G, 1 ≤ e ≤ E (4.4.44)
The sequence

(z ε h , π ε h ) converges in X h × Y h to the solution (z h , π h ) ∈ X h × Y h of the problem Find (z h , π h ) ∈ X h × Y h such that (∇z h , ∇w h ) -(π h , ∇w h ) = (∇v h , ∇w h ) f or all w h ∈ X h (q h , ∇.z h ) = 0 f or all q h ∈ Y h (4.4.45) So, v h -z h 1 = sup w h ∈X h (∇(v h -z h ), ∇w h ) w h 1
Using (4.4.45) to infer that

v h -z h 1 = sup w h ∈X h (π h , ∇.w h ) w h 1
The discrete inf-sup condition gives (4.4.45). Since (π h , ∇.z h ) = 0 and combine (4.4.45) and (4.2.2) to have (π h , ∇.u) = 0 and since (q h , ∇.(uv h )) = 0 for all q h ∈ KerB * h , we obtain

v h -z h 1 ≥ β h ẑh Z h (4.4.46) Consider w h = u h -z h in
z h -v h 2 1 ≤ C π h + q 0 h u -v h 1
The infimum over all q h ∈ KerB * h implies

z h -v h 2 1 ≤ C πh Z h u -v h 1 ,
By the inequality (4.4.46),

z h -v h 1 ≤ C β h u -v h 1 which gives u -z h 1 ≤ 1 + C β h u -v h 1
z h ∈ X h according to the problem (4.4.45). The infimum over all z h ∈ X h gives inf Now, let us derive the pressure estimate. From the discrete inf-sup condition (4.4.10) applied to q h = q hp h we have

z h ∈X h u -z h 1 ≤ 1 + C β h u -v h 1 f or all v h ∈ X h
qh -ph Z h ≤ 1 β h sup v h ∈X h I((q h -p h )∇.v h ) v h 1 . (4.4.48)
Using (4.4.4),

I((q h -p h )∇.v h ) = (q h -p, ∇.v h ) + (p -p h , ∇.v h ) (4
p -p h , ∇.v h ) = ν(∇(u -u h ), ∇v h ) + ((u.∇)u, v h ) -((u h .∇)u h , v h ) - 1 2 ((∇.u h ), u h .v h ) + a|u| α u, v h ) -a|u h | α u h , v h ) f or all v h ∈ X h
Using similar calculus as in (4.2.66)-(4.2.71), the triangle inequality and (4.4.7) to get

(p -p h , ∇.v h ) ≤ [ν||u -u h || 1 + 2ν(1 -r)||u -u h || 1 + aC||u -u h || 1 ] v h 1
Injecting this result in (4.4.49) and using (4.4.46) yields to

qh -ph Z h ≤ ν(3 -2r) + aC β h u -u h 1 + C ′ β h p -q h Consequently p -ph Z h ≤ p -qh Z h + ν(3 -2r) + aC β h u -u h 1 + C ′ β h p -q h ≤ ν(3 -2r) + aC β h u -u h 1 + 1 + C ′ β h p -q h
which finish the proof.

Theorem 4.4.4. Let (u,p)∈ X × Y be the solution of (4.2.2) and (u ε h , p ε h ) be defined by (4.4.5) then

u -u ε h 1 ≤ 3 r -1 + a νr C 1 1 + C β h inf v h ∈X h u -v h 1 + C 2 νr inf q h ∈Y h p -q h + C 3 β h ε p h (4.4.50) and p -pε h Z h ≤ ν(3 -2r) + aC 1 β h 3 r -1 + a νr C 2 1 + C 3 β h inf v h ∈X h u -v h 1 + 1 + C 4 β h 3 r -1 + a νr C 1 inf q h ∈Y h p -q h + C ′′ β 2 h ε p h (4.4.51)
Proof. For the first inequality, we use the triangle inequality

u -u ε h 1 ≤ u -u h 1 + u h -u ε h 1
Using the inequalities (4.4.33) and (4.4.19). The infimum over v h ∈ X h , q h ∈ Y h implies the estimate (4.4.50). With a similar manner 1. Compute d = P Ω 0 (ȳ -η∇J(ȳ))ȳ, with ∇J is the gradient of the cost function J and η is a positive constant given by -Consider ȳ the current point and ỹ the previous point. Compute x = ȳỹ and y = ∇J(ȳ) -∇J(ỹ) . If x T y > 0, take η = x T x x T y ; elsewhere, take η as a fixed positive value.

2. Stop when d = 0 (in practice, d < ε) and ȳ is a stationary point of J on Ω 0 .

3. Set ỹ = ȳ + ξd, with ξ is defined as the step size. And go to (1) with ȳ = ỹ.

The value y = P Ω (z) is the projection of z ∈ R 4 onto Ω. It is computed by minimizing a quadratic function of the distance of z to Ω as follow

min y∈Ω 1 2 y -z 2 = min y∈Ω 1 2 z T z -z T y + 1 2 y T y (4.5.6)
which is equivalent to

min (y 1 ,y 2 ,y 3 ,y 4 ) 4 i=1 1 2 z 2 i -z i y i + 1 2 y 2 i subject to a i ≤ y i ≤ b i , i = 1, ..., 4
y 3 -y 1 ≥ d 1 , y 2 -y 4 ≥ d 2
In order to solve the two quadratic problems, we use the Karush-Kuhn-Tucker (KKT) technique: Choose a = a 1 = a 3 , and b = b 1 = b 3 . We need three constraints with the optimization problem min

(x 1 ,x 2 ) c -(l 1 x 1 + l 2 x 2 ) + 1 2 (x 2 1 + x 2 2 ) subject to x 2 ≤ b, x 1 ≥ a, x 2 -x 1 ≥ d 1 ,
where ,a, b, d 1 , c, and l i , i=1,2, are real numbers. Consider the following matrix

A =    0 -1 1 0 -1 1    , b =    -b a d 1    ,
Then the linear constraints write

Ax -v = b, v ≥ 0, where v = (v 1 , v 2 v 3 )
T is the slack variables. The optimisation problem is solvable with unique solution due to the convexity of the cost function in R 2 . The obtained solution satisfies the KKT conditions

v = Ax -b -l + x = A T λ, v ≥ 0, λ ≥ 0, v T λ = 0,
where l = (l 1 , l 2 , l 3 ), and λ = (λ 1 , λ 2 , λ 3 ) is the vector of the Lagrange multipliers associated to the three constraints above. The unique optimal solution writes

x = l + A T λ which gives    x 1 = l 1 + λ 2 -λ 3 x 2 = l 2 -λ 1 + λ 3
For computing the Lagrange multipliers λ i , we solve a LCP problem which takes the form

v = (-b + Al) + AA T λ v ≥ 0, λ ≥ 0 v T λ = 0 Remark 4.5.1.
The gradient of the cost function is computed using the formula established in section 3.

Numerical examples

The techniques for the optimal design, developped in the previous sections, are investigated. We verify the approach for the calculation of mean flow in a vertical slot fishways (VSF). There exist various approaches to design a VSF, whose objective is to dissipate the energy of the inlet jet and to create flow conditions in the pool that enable the fish to ascend in comfort circumstances. We realize three different tests of designs of VSF with I and L shaped baffle that are the most adopted for upstream passage of fish in river obstructions. The structure of a fishway is consisting of a succession of ten pools (Figure 4.4. Each pool has a length of 1.213m and a width of 0.97m. We also have two transition pools, one at the beginning and the other at the end of the channel with a length of 1.5m and a same width as the other pools.

The numerical simulation for the three tests is done with the same physical and numerical conditions. All initial and boundary conditions are taken constants. the inflow velocity is q 1 = 0.1 m 2 s -1 , the target velocity is c = 0.8 m.s -1 . The vorticity parameter is σ = 0. The viscosity is 0.01. The exterior function is f = 0. For fish passage with comfortable conditions, we put d 1 = 0.1 and d 2 = 0.05.

Vertical Slot Fishway (2-D "L" shaped baffle-oblique slots)

The set of Equations 4.2.31 is solved by the finite element method4.4.5 combined with projected gradient for the resolution of the shape optimization problem 4.5.1-4.5.5. The implementation uses the FreeFem++ software.

FreeFem++ is a partial differential equation solver that has an advanced automatic mesh generator, it uses fast algorithms such as the multi-frontal method UMFPACK with several triangular finite elements. The central pool of the computational domain that is shown in

Slot Fishway (3-D "I" shaped baffle-rectangular slots)

This numerical example performs a 3-D computational fluid dynamics model for VSF. We consider a 3-D channel of ten pools. The geometry of slots in each pool is rectangular as shown in the Figure 4.4. The existence of the slots in the fishway provides a good dissipation of energy and quit areas between pools allowing to fishes to have a rest in the pool before the begin to across to an other pool (Figure 4.4). The ensuing results present the velocity for ten pools and compare the water flow in the random and the optimal shape. As for the initial random shape (Figure 4.5), there are a circulation areas near from slots compared to the obtained optimal shape (Figure 4.6). The results are clearly observed for the central pool. Moreover, the present 3-D calculations showed that the common assumptions in VSF that the flow is 2-D is not in general valid. The following results present the velocity for ten pools and compare the water flow in the initial and the optimal shape. As for the initial structure (Figure 4.9), there are a circulation areas near from slots compared to the obtained optimal shape (Figure 4.10). We observe clearly the recirculation regions flowing in opposite directions for the initial shape compared to results in the optimal shape where the recirculation areas near from slots are reduced.

The figure 4.11 illustrates the decreasing of the cost function J in terms of the nature of the fish pass structure.

Conclusion

The design optimization is extended to the penalty porous media model and has been successfully used in hydraulic shape optimization. Based on the Navier Stokes Forchheimer model, the penalized system has been analyzed and a finite element method are considered.

Error estimates of the velocity and the pressure are derived in term of H 1 and L 2 norm respectively. 2-D and 3-D examples have been presented, and the corresponding results have demonstrated the usefulness and robustness of the approach. relax the incompressibility term with the penalty approach. Our purpose is to use a penalty method which was introduced by Courant [START_REF] Courant | Variational methods for the solution of problems of equilibrium and vibrations[END_REF] and used by Temam [START_REF] Temam | Une méthode d'approximation de la solution des équations de Navier-Stockes[END_REF] as an application for studying the Navier Stokes equations, to treat the Navier Stokes Forchheimer equations. The penalty method is to establish an approximation of the couple (u, p) solution of (5.1.1) by (u ε , p ε ) solution of the following penalized problem

                                   u ε t -ν∆u ε + (u ε .∇)u ε + 1 2 (∇.u ε )u ε + a | u ε | α u ε + ∇p ε = f in Ω, ∇.u ε + εp ε = 0 in Ω, u ε = 0 on Γ 0 , u ε = g on Γ 1 , ν ∂u ε ∂n -p ε n = 0 on Γ 2 , u ε | t=0 = u 0 in Ω. (5.1.2)
In this chapter, we take into consideration the case of Dirichlet boundary conditions u |∂Ω = 0. We prove the existence and uniqueness of the initial problem solution. We separate the penalized problem into two parts; the linearised problem and the nonlinear one. We derive an adjoint equation from the penalized problem and compute the shape gradient of the objective function. Finally we set up a time discretization and we employ it with a discrete finite element method to discretize the penalized problem and approximate the discrete perturbed equation solution.

The penalized problem model is considered to study an optimization problem related to fish pass structure, which consist to find an optimal shape of fishway structures by minimizing a cost function. The procedure 1 is to adopt a finite element method to solve the non-stationary Navier Stokes equations with damping and combine it with a gradient type algorithm called spectral projected gradient to solve the optimization problem in order to find a velocity close to a target velocity associated with fish swimming capabilities.

The chapter is organized as follow; The second section is devoted to the preliminaries useful for the analysis. A linearised problem is treated to show some error estimates. In the last part of this section we study a non linear problem to establish a general error priori estimates. The section 3 deals with the shape design. We define a shape optimization problem. We derive the adjoint system and compute the shape gradient of the cost function. The section 4 treat a time discretization by the use of a backward Euler scheme combined with a finite element method to approximate the penalized problem and derive an error estimate for the velocity and the pressure. Finally, in the last section, we Lemma 5.2.2. There exists a constant c 0 > 0 depending on Ω and such that if cε ≤ 1,

||Av|| ≤ c 0 ||A ε v||, ||A 1/2 v|| ≤ c 0 ||A 1/2 ε v|| (5.2.12)
We write some lemma of Gronwall type which will be used in the following.

y m + k m n=0 h n ≤ k m n=0 (g n y n + f n ), with k T /k n=0 g n ≤ M ∀0 ≤ m ≤ T /k. Suppose kg n < 1 ∀n and let σ = max 0≤n≤T /R (1 -kg n ) -1 , then y m + k m n=1 h n ≤ exp(σM )(B + k m n=0 f n ), ∀m ≤ T /k
We give the property of monotonicity for any mapping F given by F :

x → |x| α x (|u| α u -|v| α v, u -v) ≥ 0 (5.2.13)
The following Sobolev inequality is useful to deal with the nonlinear form F.

p ≤ ∇v , 1 ≤ p ≤ 6 (5.2.14) It is Known that if ∂Ω is smooth enough, we have for any v ∈ V ∩ H 2 (Ω) v 2 ≤ ∆v (5.2.15)
We define the following functional G(u) = P F (u). In the coming result, we derive an estimate for the velocity's second derivative, the velocity's derivative with respect to time and the pressure.

||e|| 2 L ∞ (L 2 ) + ν||e|| 2 L 2 (H 1 ) + ε||q|| 2 L 2 (L 2 )
≤ Cε which is equivalent to (5.3.5). To deduce the assertion (5.3.6), we need to introduce the auxiliary problem: for 0 < t ≤ T 0 we define (w, φ)

w s + ν∆w + ∇φ = e(s) ∀ 0 < s ≤ t,
∇.w = 0, w(t) = 0.

(5.3.7) Firstly, we derive the following inequality

ν||w|| L 2 (H 2 ) + ||∇φ|| L 2 (L 2 ) ≤ C||e|| L 2 (L 2 ) (5.3.8)
For this aim, we use the inner product of (5.3.7) with Aw, and we integrate from 0 to t, to obtain

ν||w|| L 2 (H 2 ) + ||∇w(0)|| L 2 (L 2 ) ≤ C||e|| L 2 (L 2 )
Using the projection operator P H on (5.3.7) to get

||w s || L 2 (L 2 ) ≤ C||e|| L 2 (L 2 )
We use the equation (5.3.7) again to get

||∇φ|| L 2 (L 2 ) ≤ C||e|| L 2 (L 2 )
This latter inequality completes the demonstration of (5.3.8).

We now multiply the equation (5. We integrate from 0 to t, and the fact w(t) = e(0) = 0 gives

t 0 ||e|| 2 ds = - t 0 ε(φ, p ε )ds ≤ δ||φ|| 2 L 2 (L 2 ) + C δ ε 2 ||p ε || 2 L 2 (L 2 )
with C δ is a positive constant depending only of δ. Applying (5.3.8) and choosing δ small enough to have

t 0 ||e|| 2 ds ≤ Cε 2 ||p ε || L 2 (L 2 ) ≤ Cε 2 ∀ t ∈ [0, T 0 ].
The following result is necessary for the subsequent investigations. Proof. We consider the decomposition We integrate (5.3.17) and using the Gronwall lemma to have

H 1 0 (Ω) = V ⊕ V ⊥ , where V ⊥ = {(-∆) -1 ∇q, q ∈ L 2 (Ω)} 186CHAPTER
t 0 s 2 ||e t (s)|| 2 ds + t 2 ||∇e|| 2 + εt 2 ||q(t)|| 2 ≤ Cε 2 ∀ t ∈ [0, T 0 ]
Again from the equation (5.3.3)

||q|| 2 ≤ C||∇q|| 2 -1 ≤ C(||∆e|| 2 -1 + ||e t || 2 -1 ) ≤ C(||e|| 2 1 + ||e t || 2 )
Consequently

T 0 0 s 2 ||q|| 2 ds ≤ C T 0 0 s 2 (||e|| 2 1 + ||e t || 2 )ds ≤ Cε 2
The result below describes the behavior of the linear error part.

Theorem 5.3.1. Assume the assumption (S1). Then, there exists a constant C depending on the given data such that

||e(t)|| + t 0 ||e(s)|| 2 1 ds 1 2 ≤ C √ ε ∀ t ∈ [0, T 0 ].
If we assume moreover the hypothesis (S2), then

√ t||e(t)|| + t||e(t)|| 1 + t 0 s 2 ||q|| 2 ds 1 2 ≤ Cε ∀ t ∈ [0, T 0 ].

holds.

The following subsection treats the transfer of the results that have been derived for the linear case to the non linear one.

Error estimates for the nonlinear perturbed problem

We consider the following intermediate linear equations

v t -ν∆v + ∇γ = f -B(u, u) -a|u| α u (5.3.18) ∇.v + εγ = 0, v(0) = u 0 , ( 5
≤ Cε ∀ t ∈ [0, T 0 ].
Proof. The lemma is a consequence of Lemma.5.3.1 and Theorem.5.3.1 applied to the problem (5.3.20).

We now take η = u εv, q = p εγ, and subtracting (5.

η t -ν∆η + B(u ε , u ε ) -B(u, u) + a(|u ε | α u ε -|u| α u) + ∇q = 0 (5.3.21) ∇.η + εq = 0, η(0) = 0 (5.3.22) Or B(u ε , u ε ) -B(u, u) = B(u ε , u ε -u) + B(u ε -u, u ε ) = B(u ε , ξ + η) + B(ξ + η, u ε ) (5.3.23)
We rewrite the equation (5.3.21) as

η t + A ε η + B(u ε , ξ + η) + B(ξ + η, u ε ) + a(|u ε | α u ε -|u| α u) = 0 (5.3.24)
The following theorem states a general priori estimate for the errors e = uu ε and q = pp ε ; with (u, p) are solutions of the state problem (5.1.1), and (u ε , p ε ) are solutions of the penalized problem (5.2.42).

Theorem 5.3.2. Assume that (S1) and (S2) hold. Then we have

√ t||u(t) -u ε (t)|| + √ νt||u(t) -u ε (t)|| 1 + t 0 s 2 ||p(t) -p ε (t)|| 2 d 1 2 ≤ Cε ∀ t ∈]0, T 0 ].
Proof. Multiplying (5.3.24) by A -1 ε η, we obtain

1 2 d dt ||A 1 2 ε η|| 2 + ν||η|| 2 = -b(u ε , ξ + η, A -1 ε η) -b(ξ + η, u, A -1 ε η) -a(|u ε | α u ε -|u| α u, A -1 ε η) = I 1 + I 2 + I 3
Using (5.2.7), (5.2.8), and Cauchy Schwartz's inequality we derive that

I 1 ≤ C||u ε || 2 ||ξ + η||||A -1 ε η|| 1 ≤ C||u ε || 2 ||ξ + η||||∇A -1 ε η|| ≤ C||u ε || 2 (||ξ|| + ||η||)||A -1 2 ε η|| ≤ ν 4 ||η|| 2 + C||ξ|| 2 + C||u ε || 2 2 ||A -1 2 ε η|| 2
With the same argument, we have

I 2 ≤ ν 4 ||η|| 2 + C||ξ|| 2 + C||u|| 2 2 ||A -1 2 ε η|| 2
For the third term we proceed similarly

I 3 ≤ C||u ε -u|| 2 ||A -1 2 ε η|| ≤ C(||u ε || 2 2 + ||u|| 2 2 ) + C||A -1 2 ε η|| 2
Adding the above inequalities we get

d dt ||A -1 2 ε η|| 2 ν||η|| 2 ≤ C||ξ|| 2 + C(1 + ||u ε || 2 2 + ||u|| 2 2 )||A -1 2 ε η|| 2 + C(||u ε || 2 2 + ||u|| 2 2 ) (5.3.25)
Thanks to lemma 5.2.5,

T 0 0 (||u ε || 2 2 + ||u|| 2
2 )dt ≤ C, we can apply the Gronwall lemma to (5.3.25), using lemma 2.8 to obtain

||A -1 2 ε η(t)|| 2 + ν t 0 ||η(s)|| 2 ds ≤ C t 0 ||ξ(s)|| 2 ds ≤ Cε 2 ∀ t ∈ [0, T 0 ]. (5.3.26)
We now multiply the equation (5.3.21) by tη and (5.3.22) by tq and we sum them up, then using (5.2.7) and Cauchy Schwartz's inequality one gets

1 2 d dt t||η|| 2 + νt||∇η|| 2 + εt||q|| 2 = 1 2 ||η|| 2 -t b(u ε , ξ + η, η) -t b(ξ + η, u, η) -at(|u ε | α u ε -|u| α u, η) ≤ 1 2 ||η|| 2 + Ct||u ε || 2 ||ξ + η|| 1 ||η|| + Ct||u|| 2 ||ξ + η|| 1 ||η|| + Ct||u ε -u|| 1 ||η|| ≤ 1 2 ||η|| 2 + Ct||u ε || 2 ||ξ + η|| 1 ||η|| + Ct||u|| 2 ||ξ + η|| 1 ||η|| + Ct||(u ε -v) -(u -v)|| 1 ||η|| ≤ 1 2 ||η|| 2 + Ct||u ε || 2 ||ξ + η|| 1 ||η|| + Ct||u|| 2 ||ξ + η|| 1 ||η|| + Ct||η -ξ|| 1 ||η|| ≤ 1 2 ||η|| 2 + νt 2 ||∇η|| 2 + Ct||ξ|| 2 1 + Ct(||u ε || 2 + ||u|| 2 )||η|| 2
We integrate from 0 to t and use (5.

= νt||∇η|| 2 + εt||q|| 2 -t 2 b(u ε , ξ + η, η t ) -t 2 b(ξ + η, u, η t ) -at 2 (|u ε | α u ε -|u| α u, η t ) (5.3.29)
For the first nonlinear term in the right hand side, we proceed as follow

t 2 b(u ε , ξ + η, η t ) ≤ t 2 ||u ε || 2 ||ξ + η|| 1 ||η t || ≤ t 2 6 ||η t || 2 + Ct 2 ||u ε || 2 2 (||ξ|| 2 1 + ||η|| 2 1 ) ≤ t 2 6 ||η t || 2 + Cε 2 ||u ε || 2 2 + Ct 2 ||u ε || 2 2 ||∇η|| 2
With the same manner we obtain

t 2 b(ξ + η, u, η t ) ≤ t 2 6 ||η t || 2 + Cε 2 ||u|| 2 2 + Ct 2 ||u|| 2 2 ||∇η|| 2
For the last non linear term we have

at 2 (|u ε | α u ε -|u| α u, η t ) ≤ at 2 ||η t ||||u ε -u|| 1 ≤ t 2 6 ||η t || 2 + Ct 2 η -ξ 2 1
Summing the above inequalities into (5.3.29) to get

t 2 ||η t || 2 + ν d dt t 2 ||∇η|| 2 + ε d dt t 2 ||q|| 2 ≤ νt||∇η|| 2 + εt||q|| 2 + C(ε 2 + t 2 ||∇η|| 2 )(||u ε || 2 2 + ||u|| 2 2 ) + Ct 2 η -ξ 2 1
We integrate from 0 to t, using (5.3.27) and the Gronwall lemma we obtain which end the proof of the theorem 4.2.2.

Control problem

Adjoint equation

In this section we derive an adjoint equation related to the equation (5.1.2). For simplicity we assume that f = 0. First, we define the objective function

J(Ω) = J 1 (Ω) + J 2 (Ω), (5.4.1)
where Proof. We define also the following function Let L(Ω, u ε , p ε , v, q) be a Lagrangian functional defined by L(Ω, u ε , p ε , v, q) = J(Ω) -F (Ω, u ε , p ε , v, q), (5.4.4)

J 1 (Ω) = 1 2 T 0 Ω |u ε -u d | 2 dtdx
F (Ω, u ε , p ε , v, q) =
we first derive L with respect to the state variable p in any direction p ∈ M (Ω), we obtain ∂L ∂p (Ω, u ε , p ε , v, q).p =

Shape gradient

In this subsection, we will try to express the shape gradient using the velocity method (see Céa [4]), via the state and adjoint problems.

Let Ω be a reference domain in R 2 , the perturbation of Ω by the velocity method is described as the velocity field V(t)(x) = V(t, x), ∀x ∈ Ω, t ≥ 0.

It can generate transformations

T t (V)X = x(t, X), t ≥ 0, X ∈ Ω, the flow defined by the initial value problem    dχ dt (t, X) = V(t, χ(t)), χ(0, X) = X,

(5.4.7)

with the initial value X given. Let Ω t be a perturbation domain of Ω and J(Ω) be a functional associated to Ω t . The shape derivative of the functional J(Ω t ) at Ω in the direction of the deformation field V is written as dJ(Ω; V) = lim t→0 J(Ω t ) -J(Ω) t .

If dJ(Ω; V) exists for all V ∈ C([0, T ]; (D k (R 2 )) 2 ), for small positive constant T, the functional J is called shape differentiable at Ω and its shape gradient verifies

dJ(Ω; V) = (∇J, V) ((D k (R 2 )) 2 ) ′ ×(D k (R 2 )) 2 194CHAPTER

HYDRODYNAMIC DESIGN OPTIMIZATION USING NON STATIONARY POROUS

We assume that the boundaries Γ 1 and Γ 2 are fixed in the admissible set of domains, and we define the velocity field admissible domain as follow

V ad = V ∈ C 0 (0, τ ; (C 2 (R 2 )) 2 )|V = 0 on Γ 1 ∪ Γ 2 .
The use of the velocity V for t ≥ 0, implies the transformation of the domain Ω into Ω t = T t (Ω) by the velocity method with formulation (5.4.7). Let us find an expression of the derivative of the saddle point problem j(t) with respect to t where j(t) = min L(Ω t , u ε t , p ε t , v t , q t ) with (u, p) and (v, q) are solutions of (5.1.1) and (5.4.3) in the perturbed domain Ω t , respectively. We consider the Hilbert spaces which depend on the parameter t defined by

V 0 (Ω t ) = u ε • T -1 t : u ε ∈ V 0 (Ω) V g (Ω t ) = u ε • T -1 t : u ε ∈ V g (Ω) Ŷ (Ω t ) = p ε • T -1 t : p ε ∈ Ŷ (Ω)
since T t and T -1 t are diffeomorphisms, the parametrisation do not influence j(t). And we have j(t) = min (u ε ,p ε )∈Vg(Ω)× Ŷ (Ω) max (v,q)∈V 0 (Ω)× Ŷ (Ω)

L(Ω t , u ε • T -1 t , p ε • T -1 t , v • T -1 t , q • T -1 t )
we define the following functions which depend on the parameter t

l 1 (t) = 1 2 T 0 Ωt |u ε • T -1 t -u d | 2 dx + σ 2 T 0 Ωt |curl(u ε • T -1 t )| 2 dtdx, l 2 (t) = T 0 Ωt [ν∇(u ε • T -1 t ) : ∇(v • T -1 t ) + ((u ε • T -1 t ).∇)(u ε • T -1 t ).(v • T -1 t ) + a|u ε • T -1 t | α (u ε • T -1 t ).(v • T -1 t ) -(p ε • T -1 t )∇.(v • T -1 t )]dtdx - T 0 Ωt (q • T -1 t )∇.(u ε • T -1 t )dtdx -ε T 0 Ωt (q • T -1 t )(p ε • T -1 t )dtdx
The Lagrangian functional writes We consider V(0, X) ∈ V ad , and observe that V(0, X) = V. Therefore we can derive the shape gradient using the formula (5.4.7) [(-∇u ε .V) t v + (-∇v.V) t u ε + ν∇(-∇u ε .V) : ∇v + ν∇u ε : ∇(-∇v.V) + ((-∇u ε .V).∇u ε ).v + (u ε .∇(-∇u ε .V).v + (u ε .∇u ε ).(-∇v.V) 5.5 Discrete finite element method

L(Ω t , u ε • T -1 t , p ε • T -1 t , v • T -1 t , q • T -1 t ) = l 1 (t) -l 2 (t) If Φ : [0, τ ] × R 2 → R
d dt L(Ω t , u ε • T -1 t , p ε • T -1 t , v • T -1 t , q • T -1 t )| t=0 = l ′ 1 (0) -l ′ 2 (

Preliminaries

In this section we write some results and notations which will be used in this work. We define the following forms a(u, v) = ν(A 1/2 u, A 1/2 v), a ε (u, v) = ν(A

1/2 ε u, A 1/2 ε v), c α (u, v) = (a|u| α u, v) ∀u, v ∈ X
We begin with a general regularity result, on the perturbed solution, useful for the establishment of an error bound associated with the finite elements discretization. Theorem 5.5.1. Under the assumptions (A1) and Cε ≤ 1, there is a constant k 0 > 0 such that if k 0 ∆t ≤ 1, then the following estimates hold for all 1 ≤ m ≤ N .

||A 1/
Proof. Taking (v, q) = 2(u n ε , p n ε )∆t in (5.5.3), using (5. We can establish from (5.5.3) that

d t u n ε + νA ε u n ε + B(u n ε , u n ε ) + a|u n ε | α u n ε = f (t n ) (5.5.9)
Taking the scalar product of (5.5.9) with (ν -1 d t u n ε + A ε u n ε )∆t in Y and using the relation

2(A 1/2 ε (u -v), A 1/2 ε u) = ||A 1/2 ε u|| 2 -||A 1/2 ε v|| 2 + ||A 1/2 ε (u -v)||
+ A ε u n ε )| ≤ c||A 1/2 u n ε ||||u n ε || L ∞ ||ν -1 d t u n ε + A ε u n ε || ≤ 1 6ν ||d t u n ε || 2 + ν 6 ||A ε u n ε || 2 + ν -1 c||u n ε || 2 ||A 1/2 u n ε || 2 ||A 1/2 ε u n ε || 2 Secondly |c α (u n ε , ν -1 d t u n ε + A ε u n ε )| ≤ 1 6ν ||d t u n ε || 2 + ν 6 ||A ε u n ε || 2 +ν -1 c||A 1/2 u n ε || 2 ||A 1/2 ε u n ε || 2 Finally |(f (t n ), ν -1 d t u n ε + A ε u n ε )| ≤ 1 6ν ||d t u n ε || 2 + ν 6 ||A ε u 2 ε || 2 + ν -1 c||f (t n )|| 2
Combining these inequalities with (5.5.11) to obtain (5.5.13) for all 1 ≤ m ≤ N . By using (5.2.12), (5.5.8), and (5. We take v = 2d t u n ε ∆t in (5.5.15) and we use (5.2.5) and (5.5.7) to obtain (5.5.33)

||A 1/2 ε u n ε || 2 -||A 1/2 ε u n-1 ε || 2 + 1 2 (ν -1 ||d t u n ε || 2 + ν||A ε u n ε || 2 )∆t +c α (u n ε , ν -1 d t u n ε + A ε u n ε ) ≤ d n ||A 1/2 ε u n ε || 2 ∆t + ν -1 c||f (t n )|| 2 ∆t with d n = ν -1 c(1 + ν -2 ||u n ε || 2 )||A
||d t u n ε || 2 -||d t u n-1 ε || 2 + 2ν||A
Summing these inequality from 1 to m and using Theorem 5.5.1 and (5.2.12) we get (5.5.30) In order to establish the error estimates of the finite element penalty method, we introduce the following Galerkin projection R h ; (X, M ) → X h , Q h : (X, M ) → M h defined by a(u -R h (u, p), v h ) -d(v h , p -Q h (u, p)) + d(u -R h (u, p), q h ) + ε(p -Q h (u, p), q h ) = 0 ∀(v h , q h ) ∈ X h × M h , (5.5.34) for all (u, p) ∈ X × M with divu + εp = 0.

Numerical results

In this subsection we present some results about the optimization of a fishway using finite element method investigated in the previous sections to solve the state problem combined with a projected gradient algorithm for the resolution of the optimization problem. The domain fluid is considered as a rectangular channel consisting of ten pools. Inside each The following results represent the velocity for ten pools in the initial random shape (Figure 5.8) and the optimal shape (Figure 5.9). The nature of the studied structure takes part of the minimization of turbulence by the use of the form L of baffles which are considered as a break waves of water. The velocity is almost close from the uniform target velocity u d for the optimal shape We give numerical results for the central pool in the ensuing figures. Turbulence appears clearly in the areas near from slots in the figure 5.10 (left). For the optimal shape (Figure 5.10(right)), the nature of velocity permits a suitable passage for fish and provides for them a rest in the zones where the velocity is almost null.

Conclusion

In this work, we have studied the Navier Stokes Forchheimer equations. A penalty method has been introduced to examine different properties of a penalized problem along two parts the first one is the derivation of an error estimation and use it for the global error

  nl = ((T -2I N )U ) ∂ x U ; S (5) nl = ((T -I N )U ) U ∂ x h h ; S

0.0. 22 ) 1 t 0 U 1 .

 22101 Après avoir établi les estimations des sources termes, nous entamons l'étude du système linéarisé∂ t U -ν∂ xx U = S( Ũ , h, ∂ x Ũ , ∂ x h, ∂ xx h) := S ∂ t h + ū∂ x h = F (U, h) := F := -∂ x (ū) h (0.0.23)avec Ũ , et h sont connues. Nous divisons l'étude du système (0.0.23) en deux systèmes; l'un parabolique∂ t U -ν∂ xx U := S U (0, .) = U 0 (.) ∈ H 2 (R) (Par) et l'autre hyperbolique ∂ t h+ ū∂ x h = F h(0, .) = h 0 ∈ H 2 (R)(Hyp)Nous montrons les résultats suivants Proposition 0.0.2. Si S ∈ C(0, T, H 1 (R)) pour T > 0, alors 1. le problème (Par) admet une solution forte U vérifiantU ∈ C(0, T ; H 1 (R)) ∩ C 1 (0, T ; L 2 (R)) ∩ L 2 (0, T ; H 3 (R)) (0.0.24) 2. il existe deux constantes positives C 1 et C 2 , qui peuvent dépendre de la viscosité, tel que pour tout t ∈ [0,T]: U (t, .) 2 2 +C (τ, .) 2 3 dτ ≤ e t U 0 Soit T > 0 tel que F ∈ C(0, T ; H 1

Figure 2 . 1 :

 21 Figure 2.1: Fishway and ground plan Ω: each pool is designed by dashed lines

Figure 2 . 2 :

 22 Figure 2.2: Prototype geometry: details of slot and pool

R 2 ξ

 2 α p(ξ)e ixξ dξ where D α = D α 1 1 ....D αn n , and D j = 1 i ∂ ∂x j . We note the integral above by p(D) = D α .

  (a). (Initialization). Let ȳ ∈ Ω 0 , and let ε > 0 be a positive tolerance. (b). (Search direction computation) Let d = P Ω 0 (ȳ -η∇j(ȳ))ȳ, with η is a positive constant given by -First iteration η = 1.

  (c). (Termination) If d = 0 (in practice, d 2 < ε), then stop: ȳ is a stationary point of j on Ω 0 . (d) . (Step-size) Calculate a value ξ ∈ (0, 1] such that j(ȳ + ξd) ≤ j(ȳ) + ξδ∇j(ȳ) T d, with δ > 0 (usually, δ ∈ [10 -4 , 10 -1 ]). For the determination of a suitable step-size ξ by an iterative way, we choose ζ as a positive constant (usually, ζ = 2), take θ 1 = j(ȳ) and θ 2 = δ∇j(ȳ) T d, and then for p = 0,1,2,... define ξ = 1 ζ p , and stop when j(ȳ + ξd) ≤ θ 1 + ξθ 2 . (e) (Update) Define ỹ = ȳ + ξd, and go to (a) with ȳ = ỹ.

Figure 2 . 3 :

 23 Figure 2.3: Geometric characteristics of a pool with "I" shaped baffle

70CHAPTER 2 .Figure 2 . 4 :

 224 Figure 2.4: Initial random velocity field in the central pool

Figure 2 . 5 :

 25 Figure 2.5: Optimal velocity field in the central pool based on gradient free algorithm

Figure 2 . 7 :

 27 Figure 2.7: Non Optimal Shape and corresponding velocity

Figure 2 . 8 :

 28 Figure 2.8: Optimal Shape based on spectral projected gradient algorithm and corresponding velocity

Figure 2 . 9 :

 29 Figure 2.9: Geometric characteristics of a pool with "L" shaped baffle

  .6.16)The numerical modeling was carried out using the same physical and numerical requirements as the previous tests. The time interval for the simulation was T = 300s. The obtained optimal points are a =(0.5521,0.1581), b = (0.7770,0.0818), c = (0.3500,0.2431).

Figure 2 . 10 :

 210 Figure 2.10: Non Optimal Shape and corresponding velocity

Figure 3 .

 3 Figure 3.14 shows the water velocity at final time of the simulation corresponding to the initial random configuration. Three different regions are distinguished: a main flow region where the maximum velocities in the pool occur and two low velocity recirculation regions.The main flow first passed the slot, then flowed back after blocking by the "L" shaped baffle. In the controlled case (Figure3.16), the circulation areas near slots are strongly reduced. The results pointed out that design is better in terms of velocity and flow pattern.

Figure 2 . 11 :

 211 Figure 2.11: Optimal Shape based on gradient free algorithm and corresponding velocity

Figure 3 . 1 :

 31 Figure 3.1: description of the multilayer division of the fluid domain.

. 7 )

 7 Finally, We introduce the kinematic viscosities ν = µ ρ . With the hydrostatic hypothesis the multilayer system in d-dimensional case, for α = 1, ..., N takes the form,

1 )

 1 In this case, we can denote the horizontal velocities u H,α by u α . If we use (3.4.1), and sum the equations (3.3.5) with respect to α = 1, ..., N , we obtain

. 6 ) 2 -

 62 The flow equation is based on the evaluation of the source term K α+ 1 K α-1 2

C 2 >

 2 0, is a given constant and Eū := sup 0≤t≤T ū 2

4 . 29 )

 429 Adding (3.4.24)-(3.4.26)-(3.4.29), the Gronwall lemma gives (3.4.28).

Figure 3 . 2 :

 32 Figure 3.2: Prototype geometry: details of ten pools

y 2 -Figure 3 . 3 :

 233 Figure 3.3: Prototype geometry: details of slot and pool

5 . 17 ) 3 . 5 . 1 .

 517351 Theorem Let (p, r) ∈ L ∞ (0, T ; H 2 (Ω)) 2 be two tests functions. The adjoint system of the equations (3.5.10) with initial and boundary conditions (3.5.11) writes

  y 5 with y(ν α ); otherwise replace y 5 with y(ν β ). Go to (f) b) If Φ(y 1 ) ≤ Φ β ≤ Φ(y 4 ) then replace y 5 with y β and go to (f)c) If Φ(y 4 ) ≤ Φ β ≤ Φ(y 5 ), then calculate φ γ = Φ(y(ν γ )). If Φ γ ≤ Φ β replace y 5 withx(ν γ ) and go to (f). Otherwise go to (e) d) if Φ(y 5 ) ≤ Φ β then calculate φ δ = Φ(y(ν δ )). If Φ δ ≤ Φ y 5 , replace y 5 with y(ν δ ) and go to (f). Otherwise go to (e) e) For j = 2, ..., 5, set y j = y 1 + 1 2 (y jy 1 )f) Resort values of Φ at each resulting vertex

1 -

 1 .6.15) Through the iterations of the appointed algorithm above, the gradient of the cost function is called and used before calling the projection function as follow a) Initialization: Let ȳ ∈ ω α , and let ε > 0 be a positive tolerance b) Search direction computation: Let d = P ωα (ȳ -η∇J α (ȳ))ȳ, where ∇J α is the gradient of the cost function and η is a positive constant given by -First iteration η = Other iterations: Let ȳ be the current point and ỹ the previous point. Calculate x = ȳ-ỹ and y = ∇J α (ȳ)-∇J α (ỹ) . Then, if x T y > 0, take η = x T x x T y ; elsewhere, take η as a fixed positive value c) Termination: If d = 0 (in practice, d 2 < ε), then stop: ȳ is a stationary point of J α on ω α d) Step-size: Calculate a value ξ ∈ (0, 1] such that J α (ȳ + ξd) ≤ J α (ȳ) + ξδ∇J α (ȳ) T d, with δ > 0 (usually, δ ∈ [10 -4 , 10 -1 ]). For the determination of a suitable step-size ξ by an iterative way, we choose ζ as a positive constant (usually, ζ = 2), take θ 1 = J α (ȳ) and θ 2 = δ∇J α (ȳ) T d, and then for p = 0, 1, 2,... define ξ = 1 ζ p , and stop when J α (ȳ + ξd) ≤ θ 1 + ξθ 2 e) Update: Define ỹ = ȳ + ξd, and go to (a) with ȳ = ỹ The value of J α is computed using the expression (3.6.10). Let y = P ωα (z) is the projection of z ∈ R 4 onto ω α . P ωα (z) is deduced by minimizing a quadratic function of the distance 122CHAPTER 3. SHAPE OPTIMIZATION OF FISHWAYS IN MULTILAYER FLOW BASED ON OPTI of z to ω α as follow

Figure 4 .Figure 3 . 4 :Figure 3 . 6 :

 43436 Figure 4.3 represents the 3D mesh for the flow over a bump. The 3D horizontal velocity module for six layers is represented in figure 3.5. The used method may to capture the shock as shown in figure 3.5.

  1595) and b N M = (0.8920; 0.0378) for the Nelder Mead algorithm (NM) and a SP G = (0.4930; 0.1646) and b SP G = (0.7102; 0.0685) for the spectral projected gradient (SPG).

Figure 3 . 7 :Figure 3 . 8 :Figure 3 . 9 :

 373839 Figure 3.7: SPG: Non Optimal Shape and corresponding velocity in the ten pools at T = 300

Figure 3 . 12 :

 312 Figure 3.12: Scheme of the first pool

Figure 3 .

 3 Figure 3.13: NM: Initial (left) and optimal (right) velocities for central pool at T=300

Figure 3 . 15 :

 315 Figure 3.15: Scheme of the first pool

  0.0305, 0.05) T for the vertical projection. For numerical simulations, the geometric parameters are d 1 = 0.1, d 2 = 0.05, The obtained optimal position points are a N M = (0.5160, 0.1663), b N M = (0.7964, 0.0437) and c N M = (0.3446, 0.2927) for Nelder mead algorithm, and a SP G = (0.4786; 0.1548), b SP G = (0.6660; 0.0892), and c SP G = (0.3448; 0.3735) for the spectral projected gradient.

  and 3.18) compared to the initial random design (Figure3.16 and 3.19).

Figure 3 .

 3 Figure 3.16: NM: Non Optimal Shape and corresponding velocity at T=300

144CHAPTER 4 .

 4 ANALYSIS AND FINITE ELEMENT APPROXIMATION OF OPTIMAL CONTROL P Proof. We use the second equation in (4.2.30) into the first equation in (4.2.31) we obtain

148CHAPTER 4 .

 4 ANALYSIS AND FINITE ELEMENT APPROXIMATION OF OPTIMAL CONTROL Pwith u d is the target velocity. It depends on the ability of fish swimming.

156CHAPTER 4 .

 4 ANALYSIS AND FINITE ELEMENT APPROXIMATION OF OPTIMAL CONTROL P Therefore the second equation of (4.4.15) is established. For the first equation of (4.4.15) we use

.4. 25 )

 25 Replacing (4.4.22)-(4.4.25) in (4.4.21), yields to

4 . 26 )

 426 Substituting in (4.4.10) and using (4.4.7) and (4.4.9)

( 4 . 4 . 47 )

 4447 Substituting the (4.4.47) in (4.4.42) gives the estimate (4.4.33).

166CHAPTER 4 .Figure 4 . 1 :

 441 Figure 4.1: Conceptual layout of a vertical slot fishway

Figure 4

 4

Figure 4 . 2 :

 42 Figure 4.2: Fish pass geometry and mesh in the central pool

Figure 4 . 3 :

 43 Figure 4.3: Initial (left) and optimal (right) velocities for central pool

Figure 4 . 4 :

 44 Figure 4.4: Configuration of central pool

Figure 4 . 5 :

 45 Figure 4.5: Velocity fields for non optimal ten pool configuration

Figure 4 . 6 :

 46 Figure 4.6: Velocity fields for optimal ten pool configuration

Figure 4 . 7 :

 47 Figure 4.7: Objective function evolution

Figure 4 .

 4 Figure 4.8: 3-D Example of Fiswhay with oblique slots and mesh in the central pool

Figure 4 . 9 :Figure 4 . 10 :

 49410 Figure 4.9: Non optimal velocity for ten pool

Figure 4 . 11 :

 411 Figure 4.11: Objective function evolution

3 . 7 )

 37 by e(s), using (5.3.3) and fact ∇.w = 0 we obtain ||e|| 2 = (w s , e) + ν(∆w, e) + (∇φ, e) = d ds (w, e) -(e s , w)ν(∇e, ∇w) + (∇φ, e) = d ds (w, e) + (∇q, w) -(φ, ∇.e) = d ds (w, e)ε(φ, p ε )

t 0 s 2 0 s 2 0 s 2

 020202 ||η t || 2 ds + νt 2 ||∇η(t)|| 2 + εt 2 ||q(t)|| 2 ≤ Cε 2 (5.3.30) Using (5.2.7), we have|| B(u ε , ξ + η)|| -1 ≤ C(||u ε || 1 ||ξ + η|| 1 ) ≤ C(||u ε || 1 ||ξ|| 1 + ||η||) || B(ξ + η, u)|| -1 ≤ C(||u|| 1 ||ξ + η|| 1 ) ≤ C(||u|| 1 ||ξ|| 1 + ||η|| 1 ) a(|u ε | α u ε -|u| α u) ≤ C(||ξ|| 1 + ||η|| 1 )From (5.3.21) and(5.3.23) we obtain∇q = -η t + ν∆η -B(u ε , ξ + η) -B(ξ + η, u)a(|u ε | α u ε -|u| α u)consequently, by applying the above estimates on the previous equation we getT 0 ||q|| 2 ds ≤ T 0 ||∇q|| 2 -1 ds ≤ Cε 2

2 T 0 Ω|u ε -u d | 2 dtdx + σ 2 T 0 Ω( 5 . 4 . 2 ) 5 . 4 . 1 .

 2020542541 , and J 2 (Ω) = σ |curl(u ε )| 2 dtdx.Then the optimization problem can be expressed asmin |curl(u ε )| 2 dtdx such that (u ε , p ε ) is solution of (5.1.2)Theorem Let (u ε , p ε ) ∈ H 2 (0, T ; Ω) × Y be solution of the penalized problem (5.1.2). Then the adjoint equation associated to the equation (5.1.2) takes the form                                   -v t -ν∆v + (∇u ε ) T .v -(u ε .∇)vu ε )v + a|u ε | α-2 (|u ε | 2 v + α(u ε .v)u ε ) +∇q = (u εu d )σ curl(curl(u ε )) in Ω × [0, T ] ∇.v + εq = 0 in Ω × [0, T ] v = 0 on Γ 0 ∪ Γ 1 × [0, T ] ν ∂v ∂n + 2(u ε .n)vnq = σcurl(u ε ).τ on Γ 2 × [0, T ]

  + ν∇u ε : ∇v + (u ε .∇)u ε .v + 1 2 (∇.u ε )u ε .v + a|u ε | α u ε .v dtdx -T 0 Ω p ε ∇.v -q∇.u εεqp ε dtdx.192CHAPTER 5. HYDRODYNAMIC DESIGN OPTIMIZATION USING NON STATIONARY POROUS

- 2 Finally

 2 + εpq dtdx, the variation p is arbitrary, we get ∇.v + εq = 0 on Ω.(5.4.5)Next, we derive L with respect of the state variable u ε in the arbitrary direction ũ ∈ V 0 (Ω) we have0 = ∂L ∂u ε (Ω, u ε , p ε , v, q).ũ, u d ).ũσcurl(u ε ).curl(ũ) dtdx v t ũ + ν∇ũ : ∇v + (ũ.∇)u ε .v + (u ε .∇)ũ.v + 1 2 (∇.ũ)u ε .v + (∇.u ε )ũ.v dtdx + T 0 Ω q∇.ũ -a|u ε | α ũ + aα|u ε | α-2 (u ε .ũ)u ε .v dtdx, = T 0 Ω u εu d )σ curl(curl(u ε )) + v t + ν∆u ε -(∇u ε ) T .v + (u ε .∇)v u ε )v -(a|u ε | α v + aα|u ε | α-2 (u ε .v)u ε -∇q .ũ dtdx + ε )τν ∂v ∂n + 2(u ε .n)vnq .ũdtdx + Ω [v.ũ] T 0 dx.Considering an arbitrary direction ũ which vanishes in neighbourhood of the boundary Γ 2 , we obtain-v t -ν∆v+(∇u) T .v-(u ε .∇)v-u ε )v+a|u| α-2 (|u ε | 2 v+α(u ε .v)u ε )+∇q = (u ε -u d )-σ curl(and an arbitrary ũ in Γ 2 givesν ∂v ∂n + 2(u.n)vnq = σcurl(u ε ).τ on Γ we obtain the adjoint equation                                    -v t -ν∆v + (∇u ε ) T .v -(u ε .∇)vu ε )v + a|u ε | α-2 (|u ε | 2 v + α(u ε .v)u ε ) +∇q = (u εu d )σ curl(curl(u ε )) in Ω × [0, T ] ∇.v + εq = 0 in Ω × [0, T ] v = 0 on Γ 0 ∪ Γ 1 × [0, T ] ν ∂v ∂n + 2(u ε .n)vnq = σcurl(u ε ).τ on Γ 2 × [0, T ]v(T ) = 0 (5.4.6)

  )∈Vg(Ωt)× Ŷ (Ωt) max (vt,qt)∈V 0 (Ωt)× Ŷ (Ωt)

  is sufficiently smooth, we have the following Hadamard formulad dt T 0 Ωt Φ(t, x)dx| t=0 = T 0 Ω ∂Φ ∂t (0, x)dx + T 0 ∂ΩΦ(0, x)v(0, X).nds(5.4.8)

(

  u d ).(-∇u ε .V)dx + σ T 0 Ω curl(u ε )curl(-∇u ε .V)dx |u εu d | 2 V.n)ds + σ 2

1 2 ∇′ 2 ([ 2

 1222 .(-∇u ε .V)u ε .v + 1 2 ∇.u ε (-∇u ε .V).v + 1 2 ∇.u ε u ε .(-∇v.V) + aα|u ε | α-2 (-∇u ε .V.u ε )(u ε .v) + a|u ε | α (-∇u ε .V).v + a|u ε | α u ε .(-∇v.V) -(∇p.V)∇.v -p∇.(-∇v.V) -q∇.(-∇u ε .V) -(∇.u ε )(-∇q.V)]dxUsing Green's formula and the condition u ε = 0 on Γ 0 , we obtainl ε t -ν∆u ε + (u ε .∇)u ε + 1 2 ∇.u ε u ε + a|u ε | α u ε + ∇p).(∇v.V)]dtdx + t -ν∆v + (∇u ε ) T .v -(u ε .∇)vu ε )v + a|u ε | α-2 (|u ε | 2 v + α(u ε .v)u ε ) -∇q].(∇u ε .V)dtdx u ε .n)vnq].(∇u ε .V)dtds -ν∇u ε : ∇v + (u ε .∇u ε ).v + a|u ε | α u ε .v -p∇.v -q∇.u ε ]V.n dtds(5.4.11) 196CHAPTER 5. HYDRODYNAMIC DESIGN OPTIMIZATION USING NON STATIONARY POROUS Replacing (5.4.10) and (5.4.11) in (5.4.9) and using the fact that (u ε , p ε ) is solution of (5.1.2) and (v, q) is solution of (5.4.3) respectively yieldsdJ(Ω; V) εu d | 2 + σ|curl(u ε )| 2 )V.n)ds + (u ε ) ∧ n))V.n dtds (5.4.12) Note that u ε = 0 and v = 0 in Γ 0 we have n.(∇u ε .V) = ∇u ε .(n⊗n).V.n = ∇u ε .n.n(V.n) = (∇.u ε )(V.n) = 0, ∀x ∈ Γ 0 (5.4.13) ∂v ∂n .(∇u ε .V) = ∇u ε .(n ⊗ n).V. n) = (∇u ε : ∇v)V.n(5.4.15) Substituting (5.4.13), (5.4.14) and (5.4.15) in (5.4.12) the shape derivative takes the formdJ(Ω; V) = |u εu d | 2 + σ 2 |curl(u ε )| 2 + ∂u ε ∂n . 2ν ∂v ∂n σcurl(u ε ) ∧ n V.n dtds(5.4.16) Consequently, the shape gradient writes ∇J = 1 2 |u εu d | 2 + σ 2 |curl(u ε )| 2 + ∂u ε ∂n . 2ν ∂v ∂n σcurl(u ε ) ∧ n n (5.4.17)

1 ε|| 2 +L

 12 5.3) and the relation2(uv, u) = ||u|| 2 -||v|| 2 + ||u -v|| 2 , ∀u, v ∈ Y (5.5.7)to have||u n ε || 2 -||u n-2ν||A 1/2 u n ε || 2 ∆t + a∆t m n=1 u n ε α+2 L α+2 ≤ 2||f (t n )||||u n ε ||∆tWe sum this inequality from 1 to m and using (5.2.9) and the Young inequality, to obtain||u m ε || 2 + ν∆t α+2 ≤ ||u 0 || 2 + ν -1 c∆t m n=1 ||f (t n )|| 2 ≤ C(5.5.8)

1 f

 1 5.12) there exists a constant C 0 > 0 such that2∆t m n=1 d n ≤ C and d m = ν -1 c(1 + ν -2 ||u m ε || 2 )||A 1/2 u m ε || 2 ≤ 1 2 C 0 (5.5.14) 200CHAPTER 5. HYDRODYNAMIC DESIGN OPTIMIZATION USING NON STATIONARY POROUS for all 1 ≤ m ≤ N.Next, using (5.5.3) to obtain(d tt u n ε , v) + a ε (d t u n ε , v) + b(d t u n ε , u n ε , v) + b(u n-1 ε , d t u n ε , v) +a (d t (|u n ε | α u n ε ) , v) t (t)dt, v , ∀v ∈ X(5.5.15)

Lemma 5 . 5 . 1 . x 1 = l 1 + λ 2 -λ 3 x 2 = l 2 -λ 1 + λ 3 v T λ = 0 Remark 5 . 6 . 1 .

 5511132230561 Under the assumptions (A1), (A2), c 0 ε ≤ 1 and k 0 ∆t ≤ 1, the Galerkin projection (R h , Q h ) verifies• ||u -R h (u, p)|| +h||A 1/2 (u -R h (u, p))|| + h||p -Q h (u, p)|| ≤ c ν h(||A 1/2 u|| + ||p||), ∀(u, p) ∈ X × M (5.5.35) • ||u -R h (u, p)|| +h||A 1/2 (u -R h (u, p))|| + h||p -Q h (u, p)|| ≤ C ν h 2 (||Au|| + ||p|| 1 ), ∀(u, p) ∈ D(A) × H 1 (Ω) ∩ M (5.5.36)Then the linear constraints writeAxv = b, v ≥ 0, where v = (v 1 , v 2 v 3 )T is the slack variables. The optimisation problem has an unique solution due to the convexity of the cost function in R 2 . This solution satisfies the KKT conditionsv = Axb -l + x = A T λ, v ≥ 0, λ ≥ 0, v T λ = 0,where l = (l 1 , l 2 , l 3 ), and λ = (λ 1 , λ 2 , λ 3 ) is the vector of the Lagrange multipliers associated to the three constraints above. The unique optimal solution writesx = l + A T λ which gives  For computing the Lagrange multipliers λ i , we solve a LCP problem which takes the formv = (-b + Al) + AA T λ v ≥ 0, λ ≥ 0The gradient of the cost function is computed using the formula established in section 3.

Figure 5 . 1 :

 51 Figure 5.1: Schematic description of fishway structure

Figure 5 . 2 :

 52 Figure 5.2: Fish pass geometry L shaped baffle (left) and I shaped baffle (right) for the central pool

Figure 5 . 3 :

 53 Figure 5.3: Configuration of central pool

Figure 5 . 4 :Figure 5 . 5 :Figure 5 . 6 :

 545556 Figure 5.4: Initial and random shape and corresponding velocity for ten pools

Figure 5 . 7 : 3 -

 573 Figure 5.7: 3-D example of fishway with porous medium (left), Mesh of the central pool (right)

Figure 5 . 8 :

 58 Figure 5.8: Non optimal shape and corresponding velocity for ten pools

Figure 5 . 10 :

 510 Figure 5.10: Non optimal shape and corresponding velocity for the central pool

  [START_REF] Luigi | On the construction of suitable weak solutions to the 3d Navier Stokes equations in a bounded domain by an artificial compressibility method[END_REF] où u ε est solution du problème (0.0.5) et u d est une vitesse cible dépendante des capacités de nages des poissons et de leurs natures. σ est un paramètre de vorticité. Le système adjoint associé au problème (1.0.6) et la fonction coût (0.0.48) s'écrivent

  [START_REF] Godlewski | Numerical Approximation of Hyperbolic Systems of Conservation Laws[END_REF] De manière analogue que pour le problème continu, nous démontrons l'existence de la solution du problème discret et perturbé (0.0.51). Ensuite, nous établissons une estimation d'erreur en terme de la norme H 1 et L 2 pour la vitesse et la pression respectivement.

	h , p ε h ) définis Théorème 0.0.8. Soit (u,p)∈ X × Y solution du problème (0.0.5) et (u ε par (0.0.51) alors
	u

  La deuxième section est divisée en deux parties. Dans la première, nous étudions le cas linéaire des équations de Navier Stokes Forchheimer non stationnaires et nous dérivons une estimations d'erreur qui sera utilisée dans la deuxième partie pour le cas non linéaire.

[START_REF] Jean | Théorie du mouvement non permanent des eaux, avec application aux crues des rivières et a l'introduction de marées dans leurs lits[END_REF]

et X h , Y h sont les espace d'éléments finis associés à (X,Y). Dans la dernière section de ce chapitre, nous adoptons une technique de type gradient pour résoudre notre problème d'optimisation et minimiser la fonction coût (0.0.48) combinée à une méthode d'éléments finis discrète pour approcher le problème (0.0.5).

Le quatrième chapitre se rapporte à l'étude des équations de Navier-Stokes Forchheimer non stationnaires (0.0.7). Il est divisé en plusieurs sections. Dans la première section, nous nous préoccupons de certaines résultats utilisées pour la suite du chapitre. Nous donnons la formulation mathématique du problème puis nous mettons en place quelques propriétés de régularité sur la fonction f et la pression p.

  Weak solution with discontinuities . . . . . . . . . . . . . . . . . . . 3.2.3 Vertical velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Hydrostatic pressure multilayer model . . . . . . . . . . . . . . . . . . . .

	Contents
	1 General introduction
	2 Optimum Hydrodynamic Design using Shallow Water Equations

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Mathematical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Adjoint system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Shape derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Analysis of the adjoint problem . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Pseudo-differential tools . . . . . . . . . . . . . . . . . . . . . . . . 2.5.2 Main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.3 Existence and Uniqueness using the pseudo-differential calculus . . 2.6 Numerical methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.1 Finite volume method . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.2 A gradient free algorithm . . . . . . . . . . . . . . . . . . . . . . . 2.6.3 Spectral projected gradient . . . . . . . . . . . . . . . . . . . . . . . 2.6.4 Design examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Shape optimization of fishways in multilayer flow based on optimal control theory 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Multilayer approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 3.4 Analysis of the 2D multilayer system . . . . . . . . . . . . . . . . . . . . . 3.4.1 Mass conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Momentum conservation . . . . . . . . . . . . . . . . . . . . . . . . 3.4.3 Existence and uniqueness of solution model . . . . . . . . . . . . .

  is the kinematic pressure at the interface Γ α+1/2 (t), and p S represents the free surface pressure. The pressure variable is written in terms of layer depth in each layer α.

			3.1)
	with	N
	p α+1/2 (t, x) = p S (t, x) + ρg	h β (t, x)	(3.3.2)
		β=α+1
	where p α+1/2

Remark 3.3.1. The unknowns of the model are the layer depth h α , and the velocities u α . Definition 3.3.1. The weak formulation of (3.2.1) in Ω α (t) writes for all ϕ ∈ L 2 (Ω α (t)) and for all

  C 4 . The estimations (3.4.31) result from inequalities (3.4.28) and

	with C 5 = C
									ū, h) 2 t		0	t	U (τ, .) 2 3 dτ	1/2	(3.4.33)
						2 t	U 0	2 +	0	t	S(τ, .) 2 1 dτ	1/2	(3.4.34)
	and
	0	t	U (τ, .) 2 3 dτ	1/2	≤ C	′ 1 e C ′ 2 t		U 0	2 +		0	t		S(τ, .) 2 1 dτ	1/2	(3.4.35)
											U 0	2 +	0	t	1 dτ S(τ, .) 2	1/2
														(3.4.36)

with C 4 is a constant depending on C B and C(k). The inequality

(3.4.21)

gives the existence of two constant depending only on the viscosity

C ′ 1 , C ′ 2 , such that U (t, .) 2 ≤ C ′ 1 e C ′ Combining inequalities (3.

4.34), (3.4.35) and (3.4.33) yields to t 0 e -C(k)E ū, hτ F (τ, .) k dτ ≤ C ′ 1 e C 5 (1+E ū, h) 2 t ′ 2 +

  3. SHAPE OPTIMIZATION OF FISHWAYS IN MULTILAYER FLOW BASED ON OPTI

	h(t, x) ≥ η 0 > 0,
	and
			t	1/2
	max	(U, h)(t, .) 2,2	0	U (τ, .) 2 3 dτ	≤ E 0

  .2.20) 138CHAPTER 4. ANALYSIS AND FINITE ELEMENT APPROXIMATION OF OPTIMAL CONTROL P Existence of the pressure field: Supposing that Ω is locally Lipschitzian, thanks to lemma (4.2.2): there exists p ∈ Y satisfying (4.2.3) and As p ∈ Y and satisfies (4.2.21), the problem (4.2.22) has a solution thanks to the theorem 3.3.1 in [22]. Setting w = W in (4.2.6) to obtain ||p|| 2 ≤ C ||f || X ′ + ||u|| 1 + ||u|| L α+2 + ||u|| 2 1 ||p|| (4.2.23)

	pdx = 0	(4.2.21)
	Ω
	Consider the problem
	∇.W = p
	W ∈ X	(4.2.22)
	||W || 1 ≤ ||p||

Lemma 4.2.4. Consider u, v ∈ X such that (4.2.20) holds for u and v ,then

  .2.62) 146CHAPTER 4. ANALYSIS AND FINITE ELEMENT APPROXIMATION OF OPTIMAL CONTROL P Replacing the previous result in (4.2.61) yields to

  We assume again that the condition (4.2.46) is satisfied by f s

	154CHAPTER 4. ANALYSIS AND FINITE ELEMENT APPROXIMATION OF OPTIMAL CONTROL P
	.4.6)

  Theorem 4.4.2. Consider that u ε h is solution of the problem (4.4.5) in X h and the pressure p ε h defined by (4.4.6) in Y h . Then (i) The approximate discrete pressure p ε h is uniformly bounded in ε.

  Substituting(4.4.39) in(4.4.37) and thanks to (4.4.7), the fact that (u h ) is bounded and the Cauchy Schwartz inequality, we have

	160CHAPTER 4. ANALYSIS AND FINITE ELEMENT APPROXIMATION OF OPTIMAL CONTROL P
	1	(4.4.39)

  Lemma 5.3.2. Provided (S1) and (S2) are satisfied, we have|| 2 ds ≤ C ∀ t ∈ [0, T 0 ]. ||e t || 2 + νt 2 ||∇e t || 2 + εt 2 ||q t || 2 = t||e t || 2 + εt 2 (p t , q t )The results follows from the previous inequality and (5.2.41).

				t
	0 t Proof. We derive one time with respect to t on (5.3.3) and (5.3.4), we get s 2 ||p ε
				e tt -ν∆e t + ∇q t = 0	(5.3.9)
				∇.e t + εq t = εp t , e(0) = 0	(5.3.10)
	Multiplying (5.3.3) by te t and (5.3.10) by tq to get
	t||e|| 2 +	ν 2	d dt	t||∇e|| 2 +	ε 2	d dt	t||q|| 2 =	ν 2	||∇e|| 2 +	ε 2	||q|| 2 + εt(p t , q)
	≤ ||p t || 1 ν 2 ||∇e|| 2 + ε||q|| 2 + εt 2 2 2 d dt t 2 ≤ t||e t || 2 + εt 2 2 ||q t || 2 + εt 2 2 ||p t || 2
	Integrating over [0, t], using (5.3.11) and (5.2.41) we get
	t
	ε s 2 ||q Lemma 5.3.3. We suppose that (S1) and (S2) are satisfied then 0
			t					t
	t||e(t)|| 2 + ν	0	s||∇e(s)|| 2 ds + ε	0	s||q(s)|| 2 ds ≤ Cε 2 ∀ t ∈ [0, T 0 ],	(5.3.12)
	t 2 ||∇e(t)|| 2 +

2

Integrating from 0 to t, using Lemma 5.3.6, (5.2.41) and Gronwall lemma, we derive t 0 s||e t || 2 ds + t||e|| 2 1 + εt||q|| 2 ≤ Cε (5.3.11) We Multiply (5.3.9) by t 2 e t and (5.3.10) by t 2 q t and we sum up we obtain t || 2 ds ≤ C t 0 s||e t || 2 ds + Cε t 0 s 2 ||p t || 2 ds ≤ Cε t 0 s 2 ||q(s)|| 2 ds ≤ Cε 2 ∀ t ∈ [0, T 0 ]. (5.3.13)

  5. HYDRODYNAMIC DESIGN OPTIMIZATION USING NON STATIONARY POROUSand v = (-∆) -1 ∇q if and only if -∆v = ∇q and v| ∂Ω = 0. We note that for p(t) ∈L 2 (Ω)/R there exists a unique ϕ(t) ∈ V ⊥ such that ∇.ϕ(t) = p(t) with ϕ(t) 1 ≤ C||p(t)|| ∀ t ∈ [0, T 0 ]. (5.3.14) Moreover, if p t (t) ∈ L 2 (Ω)/R, we then have ∇.ϕ t (t) = p t (t) such that ϕ t (t) 1 ≤ C||p t (t)|| ∀ t ∈ [0, T 0 ]. (p t , q) = εt 2 (∇.ϕ t , q) = -εt 2 (ϕ t , ∇q) = εt 2 (e t , ϕ t ) + εt 2 (∇e, ∇ϕ t ) || 2 + cε 2 t 2 ||ϕ t || 2 + t 2 ||∇e|| 2 + ε 2 t 2 ||∇ϕ t || 2

								(5.3.15)
	Multiplying (5.3.3) by te and (5.3.4) by tq, we sum up and use (5.3.3), we infer
	1 2	d dt	t||e|| 2 +tν||∇e|| 2 + εt||q|| 2 =	1 2	||e|| 2 + εt(p, q) =	1 2	||e|| 2 + εt(∇.ϕ, q)
			=	1 2	||e|| 2 -εt(∇q, ϕ) =	1 2	||e|| 2 + εt(e t , ϕ) + ενt(∇e, ∇ϕ)	(5.3.16)
	= By the assumption (S1), we have 1 2 ||e|| 2 + ε d dt t(e, ϕ) -ε(e, ϕ) -εt(e, ϕ t ) + ενt(∇e, ∇ϕ) √ tp ∈ L t 2 ||e t || 2 + 1 2 d dt t 2 ||∇e|| 2 + ε 2 d dt t 2 ||q|| 2 = t||e|| 2 1 + εt||q|| 2 + εt 2 (p t , q)	(5.3.17)
	Using equation (5.3.3) and the inequality (5.3.15) to obtain
			εt 2 ≤ ||e t ≤ t 2 2 t 2 2 ||e

∞ (0, T 0 ; H 1 (Ω)/R). Using

(5.3.14)

we obtain εt(e(t), ϕ(t)) ≤ t 4 ||e(t)|| 2 + ε 2 t||ϕ|| 2 ≤ t 4 ||e(t)|| 2 + Cε 2 By integrating (5.3.16) from 0 to t, using the previous inequality, the Cauchy-Schwartz inequality, Lemma 5.3.6, (5.2.41) and (5.3.15) we deduce t||e(t)|| 2 + t 0 (νs||∇e(s)|| 2 + εs||q(s)|| 2 )ds ≤ Cε 2 + C t 0 ||e(s)|| 2 ds + cε 2 t 0 ||ϕ(s)|| 2 ds + Cε 2 t 0 s 2 ||ϕ t (s)|| 2 ds ≤ Cε 2 + Cε 2 t 0 (||p|| 2 + s 2 ||p t || 2 )ds ≤ Cε 2 Taking the inner product of (5.3.3) with t 2 e t , (5.3.10) with t 2 q we deduct t || 2 + t 2 ||∇e|| 2 + Cε 2 t 2 ||p t || 2

  5. HYDRODYNAMIC DESIGN OPTIMIZATION USING NON STATIONARY POROUS Lemma 5.3.4. We suppose that (S1) and (S2) are valid. Then we have

	with u is solution of the Navier-Stokes equation with damping (5.1.1). Taking ξ = v -u, φ = γ -p, and subtracting (5.3.18)-(5.3.19) from (5.1.1), we get ξ t 0 ||ξ(s)|| 2 ds 1 2 + √ t||ξ(t)|| + t||ξ(t)|| 1 + 1 t 0 s 2 ||φ(s)|| 2 ds 2	.3.19)

t -ν∆ξ + ∇φ = 0 ∇.ξ + εφ = -εp, ξ(0) = 0. (5.3.20)

The next Lemma describes the control of the errors through the non-linearities. 188CHAPTER

  190CHAPTER 5. HYDRODYNAMIC DESIGN OPTIMIZATION USING NON STATIONARY POROUS Multiplying the equation (5.3.21) by t 2 η t , and (5.3.28) by t 2 q, summing them up to get t 2 ||η t || 2 +

	ν 2	d dt	t 2 ||∇η|| 2 +	ε 2	d dt	t 2 ||q|| 2
						3.26), Lemma 2.8 and the Gronwall lemma we get
		t||η(t)|| 2 + ν

t 0 s||∇η(s)|| 2 ds + ε t 0 s||q(s)|| 2 ds ≤ Cε 2 (5.3.27) We now derive one time with respect to the equation (5.3.22) ∇.η t + εq t = 0 (5.3.28)

  || 2 + ||A ε u n ε || 2 + ||p n ε || 2 + a||u n ε || α+2 L α+2 ) ≤ C,(5.5.5)||d t u m ε || 2 + ||A ε u m ε || 2 + ||p m ε || 2 1 + ∆t

		m
	2 ε u m ε || 2 + ∆t	n=1	(||d t u n ε m
			n=1	||A 1/2 ε d t u n ε || 2 ≤ C,	(5.5.6)

  2 , ∀u, v ∈ X (5.5.10)|| 2 + ν -1 ||d t u n ε || 2 ∆t + ν||A ε u n ε || 2 ∆t +c α (u n ε , ν -1 d t u n ε + A ε u n ε )∆t + b(u n ε , u n ε , ν -1 d t u n ε + A ε u n ε )∆t = (f (t n ), ν -1 d t u n ε + A ε u n ε )∆t

	Firstly
		| b(u n ε , u n ε , ν -1 d t u n ε
	we obtain
	||A	1/2 ε u n ε || 2 -||A 1/2 ε u n-1 ε
		(5.5.11)
	Using (5.2.11)-(5.2.12), we get

  1/2 u n ε || 2 . Summing this inequality from 1 to m and observingthat A -1 ||d t u n ε || 2 + ν||A ε u n ε || 2 ) + ∆t -1 ||d t u n ε || 2 +ν||A ε u n ε || 2 )+∆t

			1/2 ε u 0 = A 1/2 u 0 , we get
	||A 1/2 ε u m ε || 2 +	1 2	∆t	n=1 m	n=1 (ν n	c α (u n ε , ν -1 d t u n ε + A ε u n ε )
							m	m
			≤ ||A 1/2 u 0 || 2 + ∆t	n=1	d n ||A 1/2 ε u n ε || 2 + ν -1 c∆t	n=1	||f (t n )|| 2
							m
	≤ C + ∆t If we take ∆t such that d n ∆t ≤ d n ||A 1/2 ε u n ε || 2 n=1 1 2 , by applying lemma 2.4 to (5.5.12) we infer	(5.5.12)
	||A 1/2 ε u m ε || 2 +	1 2	∆t	m n=1	(ν

  1/2 ε d t u n ε || 2 ∆t + 2 b(d t u n ε , u n ε , d t u n ε )∆t +2a (d t (|u n ε | α u n ε ) , d t u n ε ) ∆t ≤ 2 Since 2a(d t (|u n ε | α u n ε ), d t u n ε ) ≥ 0 thanks to (5.2.13). We use (5.2.9) and (5.2.12) to get Firstly2| b(d t u n ε , u n ε , d t u n ε )| ≤ c||d t u n ε ||||A 1/2 d t u n ε ||0||A 1/2 u n ε || +c||d t u n ε || 1/2 ||A 1/2 d t u n ε || 3/2 ||A 1/2 u ε || 1/2 ||u n ε || 1/2 ≤ ν 3 ||A ε d t u n ε || 2 + d n ||d t u n ε || 2 SecondlyAdding these inequalities and replacing them in (5.5.16) we have||d t u n ε || 2 -||d t u n-1 ε || 2 + ν||A 1/2 ε d t u n ε || 2 ∆t ≤ (ν -1 c + d n )||d t u n ε || 2 ∆t + ν -1 cSumming this inequality from 1 to m and using (5.5.4), (5.5.13) and (5.5.14) we obtain||d t u n ε || 2 + ν∆tFinally, we can establish from (5.5.3), (5.5.15) and the inf sup condition [?] thatν||A ε u n ε || ≤ ||d t u n ε || + || B(u n ε , u n ε )|| + |||u n ε | α u n 2| b(u n ε , u n εh , d t u n εh )| ≤ c||Au n ε ||||A 1/2 u n εh ||||d t u n εh || ≤ 1 4 ||d t u n εh || 2 + c||Au n ε || 2 ||A 1/2 u n εh || 2 2| b(u n εhu n ε , u n εh , d t u n εh )| ≤ c||u n εhu n ε || L 4 ||u n εh || L 4 ||A 1/2 d t u n εh || +c||A 1/2 (u n εhu n ε )|| 2 ||u n εh || L 4 ||A 1/2 d t u n εh || L 4 ≤ 1 4 ||d t u n εh || 2 + ch -2 ||A 1/2 (u n εu n εh )|| 2 ||u n εh || 2 +ch -2 ||u n εu n εh || 2 ||A 1/2 u n εh || 2 2|c α (u n εh , d t u n εh )| ≤ 1 4 ||d t u n εh || 2 + c||A 1/2 u n εh || 2 2|(f (t n ), d t u n εh )| ≤ 1 4 ||d t u n εh || 2 + c||f (t n )|| 2Replacing these inequalities in(5.5.32). with some sample calculation we obtainν||A 1/2 u n εh || 2 -ν||A 1/2 u n-1 εh || 2 + ||d t u n εh || 2 ∆t + ε(||p n εh || 2 -||p n-1 εh || 2 ) ≤ c||Au n ε || 2 ||A 1/2 u n εh || 2 ∆t + ch -2 ||A 1/2 (u n εu n εh )|| 2 ||u n εh || 2 ∆t + ||u n εu n εh || 2 ||A 1/2 u n εh ||∆t + c||A 1/2 u n εh || 2 + c||f (t n )|| 2

							tn t n-1	ε f t (t)dt, d t u n	(5.5.16)
	|	tn t n-1	f t (t)dt, d t u n ε	| ≤	ν 3	||A ε d t u n ε || 2 + ν -1 c	tn t n-1	||f (t)|| 2 dt
								tn t n-1	||f (t)|| 2 dt	(5.5.17)
				m
				n=1	||A 1/2 ε d t u n ε || 2 ≤ C, ∀1 ≤ m ≤ N	(5.5.18)

ε || + ||f (t n )|| (5.5.19) ||p n ε || 1 ≤ c||d t u n ε || + νc||Au n ε || + c|| B(u n ε , u n ε )|| + c|||u n ε | α u n ε || + c||f (t n )||

(

5.5.20) Applying (5.2.11)-(5.2.12), we get

Remerciements

estimates found in the second part. A finite element method has been used to discretize the fluid domain and derive an error estimate for the velocity and the pressure in term of norm H 1 and L 2 respectively. Numerical results were presented for Two 3-D examples for fishways structures. The obtained results stated the robustness and efficiency of the adopted technique.

Je désire en outre remercier les membres reconnaissances à Mr Christian DOGBE qui m'a donné la chance d

Chapter 4

Analysis and finite element approximation of optimal control problem based on porous media model

Introduction

The shape optimisation problem consists to find an optimal shape in an admissible set. It is a branch of the control domain by the partial differential equations. The purpose of this chapter is the study of different proprieties for the porous media equations based on the Navier Stokes equations with damping. We begin by showing the existence and uniqueness of the solution for the following boundary problem

where Ω ∈ R d (d=2, 3) is the fluid domain; Γ 0 is the lateral boundary, Γ 1 is inflow, top and bottom flow boundaries, and Γ 2 is the outflow boundary; u is the velocity; p is the pressure; ν is the viscosity; n is the unit normal vector to the boundary ∂Ω. α ∈ [0, 2] and a > 0 are real numbers. We introduce a perturbed problem related to the state system (4.1.1) as perturbed problem and the derivation of an error estimates to prove the convergence of the penalized solution. the section 3 is devoted to optimal control through which we will minimize a cost functional to obtain an optimal shape design of the fishways structures. A penalty finite element method approximation is introduced. Then we derive an error estimates and approximate the solution of the problem by discrete penalized solution. Finally in the last section, we adopt a gradient type algorithm with some numerical examples for 2D and 3D cases with structures already used for fishway modelling [START_REF] Alvarez-Vazquez | An optimal shape problem related to the realistic design of river fishways[END_REF], [START_REF] Chorda | Twodimensional free surface flow numerical model for vertical slot fishways[END_REF].

Mathematical model

Preliminaries

We restrict the study of the system (4.1.1) to the case g = 0 along this section. The generalized case is treated in the introduction section and along study of the control problem based on the use of the penalized problem.

For the rest of the chapter, we denote by ||.|| the L 2 -norm and by ||.|| k the norm of H k (Ω).

And we define the following Sobolev spaces X = (H 1 0 (Ω)) d , Y = L 2 (Ω), Ŷ = {q ∈ H 1 (Ω)} , M = q ∈ L 2 (Ω);

We give the property of monotonicity for any mapping F given by F :

We also denote by C ∞ 0,div the set of all C ∞ real vector functions u = (u 1 , ..., u d ) (d=2, 3) with compact support in Ω such that ∇.u = 0. We begin by showing here the existence and uniqueness results which will be used in the next sections. We prove that the velocity is bounded in term of the L 2 norm. For the existence of the solution, we use a Faedo-Galerkin method.

Existence of weak solution

In this subsection, we study the system

The weak formulation of the system (4.2.2) takes the form

The next result proves the existence of the velocity and show that it is bounded in norm

There exist at least a weak solution of (4.2.2) satisfying

for some positive constant C.

For the proof, we use the coming lemmas

Shape gradient

In this subsection, we express the shape gradient using the state and adjoint systems.

Consider Ω be a reference domain in R 2 , the perturbation of Ω by the velocity method is described as the flow defined by the initial value problem

Let Ω t be a perturbation domain of Ω and J(Ω) be a functional associated to Ω t . The shape derivative of the functional J(Ω t ) at Ω in the direction of the deformation field V is written as

), for small positive constant T, the functional J is called shape differentiable at Ω and its shape gradient satisfies

We assume that the boundaries Γ 1 and Γ 2 are fixed in the admissible set of domains. And we define the velocity field admissible domain as follow

The use of the velocity V for t ≥ 0, implies the transformation of the domain Ω into Ω t = T t (Ω) by the velocity method with formulation (4.3.8).

Let us find an expression of the derivative of the saddle point problem j(t) with respect to t, where

and (u t , p t ) and (v t , q t ) are solutions of (4.3.1) and (4.3.4) in the perturbed domain Ω t , respectively. We consider the Hilbert spaces which depend on the parameter t defined by

Since T t and T -1 t are diffeomorphisms, the parametrisation do not influence j(t), and we have

For ε a small positive constant, the inequality (4.4.20) show that ||p h || is bounded.

Numerical resolution

Numerical implementation of optimization problem

In this subsection we study the approach called spectral projected gradient to solve the shape optimization problem which consists to minimize the discrete cost function defined by

The used method is based on the successive projections into a closed and convex Ω 0 subset of R 4 of all the points y = (y 1 , y 2 , y 3 , y 4 ) ∈ R 4 satisfying and for a comfortable fish passage other constraints are set up:

0.97. Then the admissible set Ω 0 is defined as

The optimization problem writes min y∈Ω 0 J(y). (4.5.5)

The optimal position y ∈ Ω 0 is found through the following algorithm using the computation of the cost function gradient and the projection functions. Let η = 1, ȳ ∈ Ω and ε > 0 be a positive tolerance.

Chapter 5

Hydrodynamic design optimization using non stationary porous media model

Introduction

In this chapter, we are interested in studying the non-stationary Navier Stokes Forchheimer equations

where

) with a sufficiently smooth boundary ∂Ω. The unknown functions here are u(x, t) = (u 1 (x, t), u 2 (x, t), u 3 (x, t)) and p = p(x, t), which stand for the velocity filed and the pressure of the flow, respectively. In damping term, α > 1 and a > 0 are two constants. The given function u 0 = u 0 (x) is the initial velocity and the constant ν > 0 represents the viscosity coefficient of the flow. The vector n is the unit normal vector to the boundary ∂Ω = Γ 0 ∪ Γ 1 ∪ Γ 2 , where Γ 0 is the lateral boundary, Γ 1 is inflow, top and bottom flow boundaries and Γ 2 is the outflow boundary.

The resolution of the problem (5.1.1) could be difficult numerically, caused by the incompressibility constraint ∇.u = 0. A popular strategy to overcome this difficulty is te use a spectral projected gradient method to solve the shape optimization problem and we show some numerical results to prove the efficiency of this technique.

Mathematical model

Preliminaries

For the upcoming section we will study the following unsteady incompressible Navier Stokes Forchheimer equations

(5.2.1)

We consider the following notations 1

In particular, we will use ||.|| to denote the norm in L 2 (Ω) and (., .) to denote the scalar product in L 2 (Ω). Now, we introduce some operators associated with the Navier Stokes equations and the perturbed problem

We can also verify by integration by parts that 1 The vector functions and vector spaces will be indicated by boldface type.

It is easy to verify that b verifies

(5.2.5)

We have also the two following inequality which will be used in the upcoming sections

(5.2.7)

We introduce

and the powers A β of A (β ∈ R) are well defined and we have

for all u,v ∈ X.

For ε sufficiently small, we have

(5.2.8)

for C a positive constant.

Lemma 5.2.1. We have the following Gagliardo-Nirenberg inequalities

where c is a general positive constant depending only on Ω.

Lemma 5.2.5. We assume that the given data u 0 and f satisfy the following regularity

Then there exists T 1 ≤ T such that the solution of the system (5.2.1)-(5.2.2) satisfies:

(5.2.16)

Proof. Take the inner product of the equation (5.2.1) with ∆u and integrate all to get 1 2

we obtain 1 2

(5.2.20)

In particular, It is equivalent to solve the following equation (by assuming C ′ ≥ 1):

Since (F ′ (u)u t )•u t is positive definite, using the inequalities (5.2.8), Hölder's and Sobolev's inequalities yields to

Using Cauchy-Schwarz inequality, estimates (5.2.15) and (5.2.22), to have:

Applying Gronwall Lemma to (5.2.24), we obtain:

Substituting the inequalities (5.2.21), (5.2.25) and (5.2.26) in (5.2.27), we obtain the pressure estimate. In the case where the initial data u(0) and f (0) satisfy some nonlocal compatibility conditions, we take t 0 > 0 and we assume that we have an initial data (u 0 , p 0 ) such that

Lemma 5.2.6. Under the same assumptions of the lemma 5.2.5, we suppose in addition that

Then, the solution of the system (5.2.1) satisfies

(5.2.28)

Proof. Deriving one time the equation (5.2.1) with respect to t, to have

Multiplying the equation (5.2.29) by ∆u t , we obtain:

Since (F ′ (u)u tt ) • u tt is positive definite, using Young's inequality, we have

Thanks to (5.2.14), (5.2.21) and (5.2.34), since H 1 (Ω) ֒→ L 4 (Ω),

We obtain the pressure estimate

In this section we focus on the existence, the uniqueness and the regularity for the penalized problem associated to porous media model (5.2.1)-(5.2.2).

The associated penalized problem to the equations (5.2.1)-(5.2.2) writes

(5.2.42)

where B(u, v) = (u.∇)v + 1 2 (∇.u)v is the modified bilinear term. Using the operator A ε , we can reformulate the perturbed system (5.2.1)-(5.2.2) as

Lemma 5.2.7. Under the assumption (S1), there exists T 2 < T and a positive constant C independent of ε such that

46)

Proof. We use the inner product of (5.2.42) with u ε and (5.2.43) with p ε to obtain

By integrating with respect to t, we get the inequality (5.2.46).

Theorem 5.2.1. Consider u, v ∈ X such that (5.2.46) holds for u and v, then

For the upcoming sections we work in [0, T 0 ] with T 0 = min(T 1 , T 2 ). For simplicity we denote by ||v|| L p (X) the norm T 0 0 ||v|| p X dt 1/p in L p (0, T 0 ; X). C is the generic constant than can take different values in different palaces.

Error Analysis

The main result of this section is stated in Theorem 5.3.2. The proof is split into two steps: the error of the linear case is given in the first step and the second one is related to the error behavior for the fully nonlinear problem.

Linearised problem

The results of this subsection will be used in the next subsection to complete the analysis of the nonlinear Navier Stokes equations. We consider the linearised Navier Stokes equations at u = 0 :

The associated perturbed problem to the system (5.3.1) takes the form

Let e = uu ε and q = pp ε . Subtracting (5.3.2) from (5. The following result summarizes the error relating to the linearized problem Lemma 5.3.1. Let assume (S1), we have

Proof. Multiplying (5.3.3) by e and (5.3.4) by q and summing up, we obtain

We integrate the previous inequality from 0 to t ≤ T 0 , and we use the fact that e(0) = 0 and lemma 5.2.5 to have

We shall make two standing assumption on the datum u 0 and f T he initial velocity u 0 (x) ∈ D(A) with divu 0 = 0 and the f orcing f unction

for some positive constant C. We recall the two weak formulations concerning the initial problem and its penalized.

f or all (v, q) ∈ X × M (5.5.2) with the initial data u(0) = u 0 and u ε (0) = u 0 respectively. Now, we define the time discretization of the penalized weak formulation (5.5.2) by the backward Euler scheme

for all (v, q) ∈ X × M , and 1 ≤ n ≤ N , where 0 < ∆t < 1 is the time step size, t n = n∆,

and d t u 0 ε satisfies the following equation

, ∀v ∈ X with divv = 0 By using the inequality (5.2.11), we infer that (5.5.19) and (5.5.20) and using (5.2.12), we obtain

(5.5.21)

(5.5.23)

Combining the inequalities (5.5.21)-(5.5.23) with (5.5.13)-(5.5.14) and (5.5.18), we end the proof of the theorem 5.5.1

Finite element penalty method

Let us consider a real positive parameter h. We denote by (X h , M h ) the finite element pair of (X, M ). It is a discretization of these domain into triangles τ . We introduce the L 2 -orthogonal projections: π h : M → M h defined by

We shall make standing the assumption for the finite element pair (X h , M h )

There exists a mapping r h : D

Moreover, we suppose the inverse inequality

Let us introduce the following finite element discretization of the formulation (5.5.3)

where u 0 εh = r h u 0 , p 0 εh = 0, and

Theorem 5.5.2. Suppose (A1) and (A2) and εc 0 ≤ 1, ∆tk 0 ≤ 1 are valid. Then the following proprieties holds

)

(5.5.30)

for all 1 ≤ m ≤ N.

Proof. In a similar manner as the estimation (5.5.8) in the Theorem 5.5.1, we can derive (5.5.29).

Next we can establish from (5.5.28) that

εh , p n εh )∆t in (5.5.31) and using (5.5.10), we get

with divu + εp = 0, where c ν is a general positive constant depending only on Ω and ν.

Proof. The Galerkin projection has its stability from (5.5.27) and (5.5.34), and we have

Using (5.5.37)- (5.5.38) and the triangular inequality, yields to

for all (u, p) ∈ X × M with divu + εp = 0.

Let consider the dual problem: find Taking (v, q) = (u-R h (u, p), p-Q h (u, p)) in (5.5.40) and (v h , q h ) = (r h Φ, π h Ψ) in (5.5.34) and using (5.5.25) and (5.5.41) we infer that

(5.5.42) The estimation (5.5.35) is a consequence of (5.5.42) and (5.5.39) . We consider (u, p) ∈ D(A) × (H 1 (Ω) ∩ M ) with divu + εp = 0. based on (5.5.24)-(5.5.25) and (5.5.34), we deduce that

Therefore, using the triangles inequality and (5.5.25) yields to

Numerical methods

In this subsection we study the approach of spectral projected gradient for solving the shape optimization problem. This method use the projection into a closed and convex Ω 0 subset of R 4 of all the points y = (y 1 , y and for a comfortable fish passage other constraints are added as

0.97. Then the admissible set Ω is defined as

The optimization problem takes the form min y∈Ω 0 J(y).

(5.6.4)

The optimal position y ∈ Ω 0 is found through the following algorithm using the computation of the cost function gradient and the projection functions. The value of J in this algorithm is given by the ensuing discrete formula

We separate the previous quadratic problem into two quadratic problems the first one for the abscesses, the second for the ordinates and we write min

A Karush-Kuhn-Tucker (KKT) technique is used to solve the two quadratic problems above. We begin by the first one and we choose a = a 1 = a 3 , and b = b 1 = b 3 . We kept only three constraints and write the optimization problem to be solved as

subject to x 2 ≤ b,

where ,a, b, d 1 , c, and l i , i=1,2, are real numbers. We choose the following matrix

Shape optimization and applications to hydraulic structures : Mathematical analysis and numerical approximation

Abstract : We are interested in the theoretical and numerical study of different flow models (shallow water system, multilayer, stationary and non stationary porous media) and their applications to the shape optimization of some hydraulic structures. We explore the well-posedness of the models and derive the adjoint equations related to each system. A penalty method is used to relax the incompressibility constraint. We express the shape gradient of the cost function in terms of the velocity value as a state variable, the adjoint variables and the unit normal vector to the boundary of the domain.

We propose a discrete finite element method to approximate the solution for the penalized problem and establish a priori estimates to prove the convergence of the approximate solution to the solution of the non perturbed problem. Error estimates for the velocity and the pressure are given.

The optimization procedure is implemented using the continuous adjoint method and the finite element method.

Keywords : Shallow water equations, Multilayer method, Porous media models, Finite element method.