Shape optimization for Navier-Stokes equations with algebraic turbulence model: Numerical analysis and computation

Transcription

1 Nečas Center for Mathematical Modeling Shape optimization for Navier-Stokes equations with algebraic turbulence model: Numerical analysis and computation Jaroslav Haslinger and Jan Stebel Preprint no Research Team 2 Mathematical Institute of the Academy of Sciences of the Czech Republic Žitná 25, Praha 1

2 Shape optimization for Navier-Stokes equations with algebraic turbulence model: Numerical analysis and computation Jaroslav Haslinger 1 and Jan Stebel 2 1 Department of Numerical Mathematics, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, Praha 8, Czech Republic, hasling@karlin.mff.cuni.cz 2 Institute of Mathematics, Academy of Sciences of the Czech Republic, Žitná 25, Praha 1, Czech Republic, stebel@math.cas.cz Abstract: We study the shape optimization problem for the paper machine headbox which distributes a mixture of water and wood fibers in the paper making process. The aim is to find a shape which a priori ensures the given velocity profile on the outlet part. The mathematical formulation leads to the optimal control problem in which the control variable is the shape of the domain representing the header, the state problem is represented by the generalized Navier-Stokes system with nontrivial boundary conditions. This paper deals with numerical aspects of the problem. Keywords: Optimal shape design, paper machine headbox, incompressible non-newtonian fluid, algebraic turbulence model 1 Introduction The first component in the paper making process is the headbox which is located at the wet end of a paper machine. The headbox shape and the fluid flow phenomena taking place there largely determine the quality of the produced paper. The first flow passage in the headbox is a dividing manifold, called the header. It is designed to distribute the fibre suspension on the wire so that the produced paper has an optimal basis weight and fibre orientation across the whole width of a paper machine. The aim is to find an optimal shape for the back wall of the header so that the outlet flow rate distribution from the headbox results in an optimal paper quality. The paper making pulp is a mixture of wood fibres, water, filler clays and various chemicals at concentration of 1% solids to 99 % water by weight. In the large-scale simulation it seems reasonable to model this complex mixture as a 1

3 single continuum, with the fluid being an incompressible liquid described by the Navier-Stokes equations. The turbulence character of the flow in the header is a desirable phenomenon in the paper making process. Typically, the input Reynolds number is about In the modelling of turbulence, one usually uses the averaging procedure, which requires additionally a closure formula for the so-called Reynolds tensor. Since the flow in the header is steady and it is expected that the geometry of the domain changes only in the part of the boundary, we use a classical algebraic model introduced by Prandtl [20], see Section 2.3. Figure 1: The header. Figure 1 shows the geometry of the header. The inlet is on the left and the so-called recirculation on the right hand side. Typically about 10 % of the fluid flows out through the recirculation. The main outlet is performed by a number (usually several hundreds or thousands) of small tubes. This fact presents a difficulty in the numerical simulation and thus the complicated geometry of the tube bank is replaced by an effective medium using an approximate homogenization technique (see [10]). This work was motivated by some previous papers: The fluid flow model which is used here has been derived and studied numerically in [10]. The shape optimization problem has also been solved numerically and the results are presented in [11], see also [12]. The above cited papers are of formal character, skipping completely existence results. The mathematical justification of the model is done in [13]. The present paper is focused on a discretization, convergence analysis and numerical realization of the shape optimization problem. The text is organized as follows. In Section 2 we present the complete model and the known existence results. An approximation of the fluid flow model and of the shape optimization problem is studied in Section 3 and 4, respectively. Finally, Section 5 describes an implementation and presents results of several model examples. 2 Description of the model In this section we define the mathematical model of the flow in the header and the shape optimization problem, and recall the main existence results. For its justification and for proofs we refer to [3]. We start by specifying the geometry of the problem. 2

4 H 1 Γ D α(x 1 ) Γ α Γ out Γ D H 2 L 1 L 2 L 3 Figure 2: Geometry of Ω(α) and parts of the boundary Ω(α). 2.1 Admissible domains Let L 1,L 2,L 3 > 0, α max H 1 H 2 α min > 0, γ > 0 be given and suppose that α U ad, where { U ad = α C 0,1 ([0,L]); α min α α max, } α [0,L1] = H 1, α [L1+L 2,L] = H 2, α γ a.e. in [0,L]. (1) Here L = L 1 + L 2 + L 3. With any function α U ad we associate the domain Ω(α), see Fig. 2: { } Ω(α) = x = (x 1,x 2 ) R 2 ;0 < x 1 < L,0 < x 2 < α(x 1 ) (2) and introduce the system of admissible domains O = { Ω; α U ad : Ω = Ω(α) }. Further we will need the domains Ω = (0,L) (0,α max ) and Ω 0 = ( (0,L 1 ) (0,H 1 ) ) ( (0,L) (0,α min ) ) ( (L 1 +L 2,L) (0,H 2 ) ). Notice that Ω 0 Ω Ω for all Ω O. Clearly Ω(α) C 0,1 for all α U ad. We will denote the parts of the boundary Ω(α) as follows (see Fig. 2): { } Γ D = x Ω(α);x 1 = 0 or x 1 = L { } Γ out = x Ω(α);L 1 x 1 L 1 + L 2,x 2 = 0 { } Γ α = x Ω(α);L 1 x 1 L 1 + L 2,x 2 = α(x 1 ) Γ f = Ω(α) \ ( Γ D Γ out Γ α ). The components Γ D, Γ out and Γ f do not depend on α U ad. 2.2 Formulation of the shape optimization problem Let Γ Γ out and v opt L 2 ( Γ) be a given function representing the desired velocity profile at the outlet. We are interested in the problem min J(α,v,p) := v 2 v opt 2 s.t. α U ad, (v,p) solves (3) (4) in Ω(α). Γ 3

5 Since Γ is fixed, it is obvious that J does not depend explicitly on α. Further we will consider only a class of weak solutions to (3) (4) which will be specified in what follows. 2.3 Classical formulation of the state problem The fluid motion in Ω(α) is described by the generalized Navier Stokes system } div T(p, D(v)) + ρdiv(v v) = 0 in Ω(α). (3) div v = 0 Here v means the velocity, p the pressure, ρ is the density of the fluid and the stress tensor T is defined by the following formulae: T(p, D(v)) = pi + 2µ( D(v) )D(v), µ( D(v) ) := µ 0 + µ t ( D(v) ) = µ 0 + ρl 2 m,α D(v), where µ 0 > 0 is a constant laminar viscosity and µ t ( D(v) ) stands for a turbulent viscosity. The function l m,α represents a mixing length in the algebraic model of turbulence and it has the following form (see [11] for more details): ( l m,α (x) = 1 ( 2 α(x 1) d ) 2 α(x) 0.06( 1 2d ) ) 4 α(x), α(x 1 ) α(x 1 ) where d α (x) = min {x 2,α(x 1 ) x 2 },x Ω(α). The equations are completed by the following boundary conditions: v = 0 on Γ f Γ α, v = v D on Γ D, v τ = v 1 = 0 on Γ out, T 22 := Tν ν = σ v 2 v 2 on Γ out, (4) where ν,τ stands for the unit normal and tangential vector to Γ out, respectively and σ > 0 is a given suction coefficient. The condition (4) 4 was suggested in [10] by a numerical study of the pipe flow. It can be also considered as a condition describing a porous wall, which can be derived from the Forchheimer equation, an analogy of the Darcy equation for high velocity flows [1, 14]. By a classical solution we mean any velocity field v (C 2 (Ω(α))) 2 (C 1 (Ω(α))) 2 and a pressure p C 1 (Ω(α)) C(Ω(α)) satisfying (3) and (4). 2.4 Weak formulation of the state problem Throughout the paper we will assume that there exists a function v 0 ( W 1,3 (Ω 0 ) ) 2 which satisfies the Dirichlet boundary conditions in the sense of traces, i.e. v 0 ΓD = v D, v 0 Ω0\(Γ D Γ out) = 0, v 0 τ Γout = 0 and, in addition, div v 0 = 0 in Ω 0. We extend v 0 by zero on Ω \ Ω 0 so that v 0 (W 1,3 ( Ω)) 2 and div v 0 = 0 in Ω (the extended function v 0 will be denoted by the same symbol). Observe that such v 0 is independent of α U ad. 4

6 The norm in the space W k,p (Ω(α)) will be denoted by k,p,ω(α) in what follows. If k = 0, then notation p,ω(α) will be used. For any α U ad we denote { V 0 (α) = ϕ = (ϕ 1,ϕ 2 ) C0 (Ω(α)) C (Ω(α)); } dist(supp(ϕ 2 ), Ω(α) Γ out ) > 0 and define the spaces for the velocity W(α) = (C (Ω(α))) 2 α, W 0 (α) = V 0 (α) α, (5) where the closure is taken in the norm Finally, let v α := v 1,2,Ω(α) + M α D(v) 3,Ω(α) + div v 3,Ω(α), M α (x) := ( l m,α (x) ) 2/3, x Ω(α). W v0 (α) = { v W(α); v v 0 W 0 (α) }. We say that v W(α) satisfies the stable boundary conditions (4) 1 3 in the weak sense iff v W v0 (α). Lemma 1. W(α) and W 0 (α) are separable reflexive Banach spaces. Definition 1. Define the operator A α : W(α) ( W(α) ) by the formula A α (v),w α := 2ρ Mα D(v) D(v) 3 : D(w); v,w W(α). Ω(α) Here, α denotes the duality pairing between ( W(α) ) and W(α). We are ready to give a weak formulation of the state problem. In what follows we will use the Einstein summation convention, i.e. a i b i := n i=1 a ib i. Further we denote (f,g) α := Ω(α) fg, provided that fg L1 (Ω(α)). Definition 2. A pair {v,p} W(α) L 3 2 (Ω(α)) is said to be a weak solution of the state problem (P(α)) iff (i) v W v0 (α); (ii) for every ϕ W 0 (α) it holds: v i 2µ 0 (D(v), D(ϕ)) α + ρ(v j,ϕ i ) α + A α (v),ϕ α x j + σ v 2 v 2 ϕ 2 (p,div ϕ) α = 0; (6) Γ out (iii) for every ψ L 3 2 (Ω(α)) it holds: (ψ,div v) α = 0. Convention. In the sections, where we will deal with the state problem on a fixed domain Ω(α), α U ad, the letter α in the argument will be often omitted. Thus we will write Ω := Ω(α), W := W(α), A := A α, (, ) := (, ) α etc. without causing confusion. 5

7 2.5 Existence of a weak solution Recall that the function v 0 is now defined in the whole Ω and it does not depend on α U ad. This fact will be used further in order to establish estimates which are independent of α U ad. Theorem 2. Let σ > ρ 2. (7) Then (i) for every α U ad there exists at least one weak solution of (P(α)); (ii) there exists a constant C E := C E (µ 0,ρ,σ, v 0 3, b Ω ) > 0 such that for any weak solution {v,p} of (P(α)), α U ad, the following estimate holds: v 2 2,Ω + M D(v) 3 3,Ω + v 2 3 3,Γ out + p ,Ω C E. (8) In addition, the constant C E does not depend on α U ad ; (iii) if {v,p 1 } and {v,p 2 } are two weak solutions of (P(α)), α U ad, then p 1 = p 2. Moreover, for v 0 3, b Ω small enough (independently of α U ad ) there exists a unique weak solution. For the proof we refer to [3]. 2.6 Existence of an optimal shape Note that the assumption of Theorem 2, which guarantees the existence of at least one weak solution to the state problem (P(α)), does not depend on a particular choice of Ω(α) O. In what follows we assume that this assumption is satisfied. Further let { Ŵ(α) := v ( W 1,2 (Ω(α)) ) } 2 ; div v L 3 (Ω(α)), M α D(v) L 3 (Ω(α)) and define { Ŵ v0 (α) := v Ŵ(α); v satisfies the Dirichlet } conditions (4) 1 (4) 3 on Ω(α). Remark 1. It holds that W v0 (α) Ŵv 0 (α). The question arises, if these spaces are identical. This is in fact the density problem. For the moment we do not know the answer. The lack of the mentioned density property leads us to modify the definition of the state problem as follows: Definition 3 (Augmented state problem ( P(α))). Let α U ad. A pair (v,p) := (v(α),p(α)) Ŵv 0 (α) L 3 2 (Ω(α)) is said to be a solution of the augmented state problem ( P(α)) iff (v,p) satisfies (ii) and (iii) of Definition 2; 6

8 (v, p) satisfies the estimate (8). Clearly any solution of (P(α)) becomes a solution of ( P(α)) too. Moreover the statement of Theorem 2 can be applied to ( P(α)) as well; in particular we have the same criterion for uniqueness. Definition 4 (Augmented shape optimization problem ( P)). Let us define the set Ĝ := {(α,v,p); α U ad, (v,p) is a solution of ( P(α))}. A triple (α,v,p ) Ĝ is said to be a solution of the augmented shape optimization problem ( P) iff J(α,v,p ) J(α,v,p) (α,v,p) Ĝ. Next we introduce convergence of a sequence of domains. Definition 5. Let {Ω(α n )}, α n U ad be a sequence of domains. We say that {Ω(α n )} converges to Ω(α), shortly Ω(α n ) Ω(α), iff α n α in [0,L]. As a direct consequence of the Arzelà Ascoli theorem we see that the system O is compact with respect to convergence introduced in Definition 5. In [3] we proved the following stability result for the solutions {(v n,p n )} of ( P(α n )). Theorem 3. Let (v(α n ),p(α n )) be solutions to ( P(α n )), n = 1,2,... and α U ad satisfy α n α in [0,L], n. Then there exists v ( W 1,2 ( Ω) ) 2, p L 3 2 ( Ω) and a subsequence of {(ṽ n, p n )} (denoted by the same symbol) such that ṽ n v in ( W 1,2 ( Ω) ) 2, M αn D(ṽ n ) M α D( v) in ( L 3 ( Ω) ) 2 2, (9) p n p in L 3 2 ( Ω), n, where the symbol stands for the zero extension of a function from the domain of its definition to Ω. In addition, denoting v(α) := v Ω(α) and p(α) := p Ω(α), then (v(α),p(α)) solves ( P(α)). Corollary 4. Problem ( P) has a solution. 3 Approximation of the flow problem In this section we describe the finite-element approximation of (P(α)) and analyze its properties such as the existence of discrete solutions and their convergence to a solution of the original problem. Let Ũad U ad be a set of all piecewise linear functions α U ad. Throughout this section we will assume that α Ũad is fixed (hence the symbol α will be often dropped), so that Ω := Ω(α) is a polygonal domain. Let {T h }, h 0+ be a family of triangulations of Ω and h be the norm of T h. Throughout the section we will assume that the following conditions are satisfied: 7

9 (A1) the family {T h } is uniformly regular with respect to h: there is θ 0 > 0 such that θ(h) θ 0 h > 0, where θ(h) is the minimal interior angle of all triangles from T h ; (A2) the family {T h } is consistent with the decomposition of Ω into Γ out and Ω \ Γ out. In what follows we will assume that W 0h W 0 and L h L 3 2 (Ω) are finite dimensional spaces. Definition 6. We say that (W 0h,L h ) satisfy the inf-sup condition (also the Babuška-Brezzi condition), if there exists a constant C BB > 0 independent of h and α Ũad s.t. (q,div w) inf sup q L h w W 0h q 3 w C BB. (10) α 2 Let us emphasize that we reqiure the constant C BB to be independent of α Ũ ad, which will be important in the shape optimization part. While in the literature there are many examples of inf-sup stable elements for the situation when velocity is prescribed on the whole boundary Ω, the choice of (W 0h,L h ) satisfying (10) may not be obvious. Denote L q 0 (Ω) := {ψ L q (Ω); Ω } ψ = 0. Lemma 5. Assume that there exists ϕ W 0h such that ϕ ν > 0. Let V h W 0h W 1,2 0 (Ω) 2, Q h L h L (Ω) be finite dimensional spaces satisfying inf q Q h Γ out (q,div w) sup w V h q 3 w C (11) α 2 with a constant C > 0 independent of h and α Ũad. Then (10) holds true. Proof. Let us pick arbitrary ϕ W 0h such that ϕ α = 1 and β := ϕ ν > 0. For any q L h we can write q = q + c, where q Q h and c R. Moreover it can be easily shown that Γ out q 3 2 q c Ω (12) From (11) we see that there exists w V h, w α = 1, such that (q,div w) C 2 q 3 2. Setting w := w + C 4 (sgn c)ϕ, using Hölder s inequality and (12) we obtain: ( q,div w) = (q,div w) + C 4 sgn c(q,div ϕ) + C 4 β c C ( ) q β c C {1, 2 4 min β Ω which completes the proof. } q 3 2, 8

10 It is possible to take usual finite element spaces V h and Q h which are inf-sup stable in the norms of W 1,3 0 (Ω) and L (Ω) (such as the Taylor-Hood elements, see e.g. [2] for particular examples). Since the norm 1,3 is stronger than α, (11) holds true. Based on Lemma 5, one can easily obtain W 0h, L h satisfying (10). Definition 7. A pair (v h,p h ) W L h is said to be a solution of the discrete state problem (P h (α)) iff (i) v h v 0 W 0h, (ii) for every ϕ h W 0h it holds: v hi 2µ 0 (D(v h ), D(ϕ h )) + ρ(v hj,ϕ hi ) + ρ x j 2 ((div v h)(v h v 0 ),ϕ h ) + ( div v h div v h,div ϕ h ) + A(v h ),ϕ h + σ v h2 v h2 ϕ h2 (iii) for every ψ h L h it holds: (ψ h,div v h ) = 0. Γ out (p h,div ϕ h ) = 0, (13) Let us point out that in contrast to (P(α)), problem (P h (α)) contains the additional terms ρ 2 ((div v h)(v h v 0 ),ϕ h ) and ( div v h div v h,div ϕ h ) in order to obtain a uniform estimate for the discrete solutions. In the continuous case, these terms vanish due to the divergence free velocity. However, (iii) of (P h (α)) does not guarantee that div v h = 0 a.e. in Ω. 3.1 Existence of a discrete solution We will use a technique that is similar to the one presented in [3], Section 2.4, to prove that (P h (α)), α Ũad possesses a solution. Theorem 6. Let σ > ρ 2 Then for every h > 0 and the Babuška-Brezzi condition (10) be satisfied. (i) there exists a solution of (P h (α)); (ii) any solution (v h,p h ) of (P h (α)) admits the estimate v h 2 2,Ω + M D(v h ) 3 3,Ω + div v h 3 3,Ω + v h2 3 3,Γ out + p h ,Ω C E, (14) where the constant C E := C E (µ 0,ρ,σ, v 0 3, b Ω ) > 0 is the same as in Theorem 2, in particular independent of h and α Ũad; (iii) for v 0 3, b Ω small (independently of h), the solution is unique; (iv) p h is uniquely determined by v h. Theorem 6 will be proven in three steps. First we will deal with the existence of v h, then for given v h we will establish p h and finally uniqueness of v h and p h will be discussed. 9

11 Proof. Let us define the mapping div h : W 0h L h as follows: div h w h,ψ h := ψ h div w h w h W 0h, ψ h L h Ω and denote V h := ker div h. We want to find v h W, such that (i) v h v 0 V h, (ii) for every ϕ h V h it holds: v hi 2µ 0 (D(v h ), D(ϕ h )) + ρ(v hj,ϕ hi ) + ρ x j 2 ((div v h)(v h v 0 ),ϕ h ) + ( div v h div v h,div ϕ h ) + A(v h ),ϕ h + σ v h2 v h2 ϕ h2 = 0. (15) It is readily seen that for ϕ h V h the equations (13) and (15) coincide. Using the technique of [3], Lemma 8, one can prove the existence of v h by means of a priori estimates and the Brouwer fixed point theorem. Moreover, the estimate v h 2 2,Ω + M D(v h ) 3 3,Ω + div v h 3 3,Ω + v h2 3 3,Γ out C E, (16) Γ out holds with a constant C E > 0 independent of h > 0 and α Ũad. Now let us define the functional B h W 0h : v hi B h,ϕ h := 2µ 0 (D(v h ), D(ϕ h )) + ρ(v hj,ϕ hi ) + ρ x j 2 ((div v h)(v h v 0 ),ϕ h ) + ( div v h div v h,div ϕ h ) + A(v h ),ϕ h + σ v h2 v h2 ϕ h2, ϕ h W 0h. Γ out (17) In virtue of (15), we see that B h (V h ). From the well known properties of linear mappings in finite dimensional spaces it follows that (V h ) = (ker div h ) = R(div h). Here V h is the polar set of V h and div h : L h W 0h is the adjoint of div h (see e.g. [8]). The last equality yields the existence of p h L h satisfying div h p h = B h, meaning that div h p h,ϕ h = (p h,div ϕ h ) = B h,ϕ h for every ϕ h W 0h. Using this, (10), (16) and (17) we obtain: p h 3 2 C, where C > 0 is a constant independent of h and α Ũad. Uniqueness of v h can be proven in the same way as in [3], Lemma 13. To prove (iv), let us assume that (v h,p 1 h ) and (v h,p 2 h ) are two solutions of (P h(α)). Then, if we insert (v h,p 1 h ) and (v h,p 2 h ) into (13) and subtract the respective equations, we obtain: From (10) it follows that p 1 h = p2 h a.e. in Ω. ϕ h W 0h (p 1 h p 2 h,div ϕ h ) = 0. (18) 10

12 3.2 Convergence of discrete solutions In this section we will study the relation between (v h,p h ) and (v,p) for h 0+. Convention. Here and in what follows we will use the same symbol for an original sequence and its subsequences. Theorem 7. Let the assumptions of Theorem 6 be satisfied and let {W 0h } h>0, {L h } h>0 be dense in W 0 and L 3 2 (Ω), respectively. Then for any sequence {(v h,p h )} of solutions to (P h (α)) there exists a subsequence and a limit pair (v,p) W v0 L 3 2 (Ω) such that v h v in W, p h p in L 3 2 (Ω), h 0+ (19a) (19b) and (v,p) is a solution of (P(α)), α Ũad. For the proof of this theorem we will need the following auxiliary result which can be established using Lemma 1.19 in [18]. Lemma 8. (Some properties of A α, α U ad ) (i) A α is monotone in W(α) in the following sense: A α (v) A α (w),v w α C M α D(v w) 3 3,Ω v,w W(α), where C > 0 is independent of α. (ii) A α is continuous in W(α). Proof of Theorem 7. The existence of v W, A W, d L 3 2 (Ω) satisfying v h v in W, (20) v h v in W 1,2 (Ω), (21) MD(v h ) MD(v) in L 3 (Ω), A(v h ) A in W, (22) div v h div v h d in L 3 2 (Ω), (23) p h p in L 3 2 (Ω), follows from (14). Next we prove that div v = 0 a.e. in Ω. Let ψ L 3 2 (Ω). Then there is a sequence {ψ h }, ψ h L h, such that ψ h ψ in L 3 2 (Ω). From this and (21) we obtain: 0 = (ψ h,div v h ) (ψ,div v), (24) so that div v = 0 a.e. in Ω. Now we make the limit passage in (13). Let ϕ W 0. Then there is a sequence {ϕ h }, ϕ h W 0h such that ϕ h ϕ in W 0. (25) 11

13 Similarly to the proof of Lemma 9 in [3], we will use the compact imbeddings in the respective spaces, (19), (24) and (25) to pass to the limit with h 0+ in the standard terms, which together with (22) and (23) yield: (D(v), D(ϕ))+(v j v i x j,ϕ i )+(d,div ϕ)+ A,ϕ + v 2 v 2 ϕ 2 (p,div ϕ) = 0 Γ out for every ϕ W 0 (here we put 2µ 0 = ρ = σ = 1 for simplicity). Now we use monotonicity of div v div v and A to show that Indeed, let ϕ W. Then (d,div ϕ) + A,ϕ = A(v),ϕ for every ϕ W. 0 ( div v h div v h div ϕ div ϕ,div(v h ϕ)) + A(v h ) A(ϕ),v h ϕ (26) v hi = (D(v h ), D(v h v 0 )) (v hj,v hi v 0i ) 1 x j 2 ((div v h) (v h v 0 ),v h v 0 ) v h2 v h2 (v h2 v 02 ) + ( div v h div v h,div(v 0 ϕ)) Γ out ( div ϕ div ϕ,div(v h ϕ)) + A(v h ),v 0 ϕ A(ϕ),v h ϕ, (27) making use of (13) and the fact that v h v 0 W 0h. Letting h 0+ and using lower semicontinuity of D(v h ) 2,Ω and continuity of the remaining terms we obtain: 0 (D(v), D(v v 0 )) (v j v i x j,v i v 0i ) v 2 v 2 (v 2 v 02 ) +(d,div(v 0 ϕ)) ( div ϕ div ϕ,div(v ϕ))+ A,v 0 ϕ A(ϕ),v ϕ. (28) From (26) and (28) we arrive at the inequality Γ out 0 (d div ϕ div ϕ,div(v ϕ)) + A A(ϕ),v ϕ, (29) which holds for any ϕ W. Choosing ϕ := v ±λψ, λ > 0, ψ W and dividing by λ we obtain for λ 0+: (d,div ψ) + A,ψ = A(v),ψ. From this and (26) we see that (v,p) solves (P(α)). To prove strong convergence of v h to v we use (i) in Lemma 8: C ( D(v h v) 2 2,Ω + M D(v h v) 3 3,Ω + div v h 3 ) 3,Ω (D(v h v), D(v h v)) + A(v h ) A(v),v h v + div v h 3 3,Ω = (D(v h ), D(v h v 0 )) + A(v h ),v h v 0 + ( div v h div v h,div(v h v 0 )) + (D(v h ), D(v 0 v)) (D(v), D(v h v)) + A(v h ),v 0 v A(v),v h v. (30) The expression (D(v h ), D(v h v 0 )) + A(v h ),v h v 0 + ( div v h div v h,div(v h v 0 )) on the right hand side of (30) can be replaced using (13). Then due to weak convergence of {v h } and {p h } the right hand side of (30) vanishes for h 0+, which yields (19a). 12

14 A 0 A 1/2 A 1 A 2 A n 1/2 Ω(s κ ) κ A 3/2 A n a 0 a 1/2 a 1 a 3/2 a 2 a n 1/2 a n Figure 3: Approximation of the boundary of Ω(α). 4 Approximation of the shape optimization problem 4.1 Parameterization of the discrete shapes We now introduce two types of discretized domains: a discrete design and discrete computational domain. The boundary Γ α of the discrete design domain is realized by a smooth, piecewise quadratic Bézier function. The optimal discrete design domain is the main output of the computational process according to which a designer makes decisions. On the other hand, our finite element method requires a polygonal computational domain. Let κ > 0 be a discretization parameter, κ : L 1 = a 0 < a 1 <... < a n = L 1 +L 2 be an equidistant partition of [L 1,L 1 +L 2 ], a i = L 1 + i n L 2, n = n(κ) = L 2 κ and a i 1/2 be the midpoint of [a i 1,a i ], i = 1,...,n. Further let A i 1/2 = (a i 1/2,α i ), α i R, i = 1,...,n be the design nodes, A i = 1 2 (A i 1/2 + A i+1/2 ) be the midpoint of the segment [A i 1/2,A i+1/2 ], i = 1,...,n 1, A 0 = (a 0,H 1 ), and A n = (a n,h 2 ), see Figure 3. We introduce the set U κ := { s κ C([0,L]); s κ [0,L1] = H 1, s κ [L1+L 2,L] = H 2, s κ [ai 1,a i] is a quadratic Bézier function } determined by {A i 1,A i 1/2,A i }, i = 1,...,n. In order to define a family of admissible shapes locally realized by Bézier functions, it is necessary to specify α i R defining the position of the design nodes A i 1/2, i = 1,...,n. With the partition κ we associate the set U R n : U = { α = (α 1,...,α n ) R n ; α min α i α max, i = 1,...,n; α i+1 α i κ γ, i = 1,...,n 1; 2 α 1 H 1 κ γ, 2 α n H 2 κ } γ, where γ > 0 is the same as in (1). The family of the admissible discrete design 13

15 domains is now represented by where O κ = {Ω(s κ ); s κ U κ ad}, U κ ad = {s κ U κ ; the design nodes A i 1/2 = (a i 1/2,α i ), i = 1,...,n, are such that α = (α 1,...,α n ) U}. Due to properties of the Bézier functions it holds that U κ ad U ad. We now turn to the definition of the computational domains. To this end we introduce another family of partitions { h }, h 0+, of [L 1,L 1 + L 2 ] (not necessarily equidistant), whose norm will be denoted by h. Next we will suppose that h 0+ iff κ 0+. Let r h s κ be the piecewise linear Lagrange interpolant of s κ U κ ad on h. The computational domain related to Ω(s κ ) will be represented by Ω(r h s κ ); i.e. the curved side Γ sκ, being the graph of s κ U κ ad, is replaced by its piecewise linear Lagrange approximation r h s κ on h. The system of computational domains will be denoted by O κh in what follows: O κh := {Ω(r h s κ ); s κ U κ ad}. Since Ω(r h s κ ) is already polygonal, one can construct its triangulation T h (s κ ) with the norm h > 0 and depending on s κ U κ ad. Convention. The domain Ω(r h s κ ) with a given triangulation T h (s κ ) will be denoted by Ω h (s κ ) in what follows. 4.2 Formulation of the discrete problem Let us define the set G κh := {(s κ,v h,p h ); s κ U κ ad, (v h,p h ) is a solution of (P h (s κ ))}. The discretization of (P) then reads as follows: Find (s κ,v h,p h ) G κh such that J(s κ,v h,p h ) J(s κ,v h,p h ) (s κ,v h,p h ) G κh. (P κh ) The approximate optimal shape is given by Ω(s κ). Next we will analyze the existence of solutions to (P κh ) and their relation to solutions of ( P) as h, κ Existence of solutions In order to establish the existence results, we have to impose additional assumptions on the family of triangulations {T h (s κ )}, h, κ 0+, which are listed below. We will suppose that, for any h, κ > 0 fixed, the system {T h (s κ )}, s κ U κ ad consists of topologically equivalent triangulations, meaning that (T1) the triangulation T h (s κ ) has the same number of nodes and the nodes still have the same neighbors for any s κ U κ ad ; 14

16 (T2) the positions of the nodes of T h (s κ ) depend solely and continuously on variations of the design nodes {A i 1/2 } n i=1. For h, κ 0+ we suppose that (T3) the family {T h (s κ )} is uniformly regular with respect to h, κ and s κ U κ ad : there is θ 0 > 0 such that θ(h,s κ ) θ 0, h, κ > 0, s κ U κ ad, where θ(h,s κ ) is the minimal interior angle of all triangles from T h (s κ ). Finally, due to the mixed boundary conditions, we suppose that (T4) the family {T h (s κ )} is consistent with the decomposition of Ω h (s κ ) into Γ out and Ω h (s κ ) \ Γ out. Let us note that (T3) (T4) imply the assumptions (A1) (A2) from the previous section. One can easily show that (P κh ) leads to the following nonlinear programming problem: min J (q(α)) subject to (α,q(α)) U Rm (P n ) R(α,q(α)) = 0, where J, R, q(α) is the algebraic representation of J, (P h (s κ )), and (v h,p h ), respectively. Remark 2. From (T1) it follows that m := m 1 + m 2, where m 1 := dim W 0h and m 2 := dim L h, does not depend on s κ Uad κ or equivalently on α U. The components of the residual vector R are given by R k (α,q(α)) := 2µ 0 (D(v h ), D(ϕ k h)) Ωh (s κ) + ρ(v hj v hi x j,ϕ k hi) Ωh (s κ) + ρ 2 ((div v h)(v h v 0 ),ϕ k h) Ωh (s κ) + ( div v h div v h,div ϕ k h) Ωh (s κ) + A(v h ),ϕ k h Ωh (s κ) + σ v h2 v h2 ϕ k h2 (p h,div ϕ k h) Ωh (s κ), k = 1,...,m 1, Γ out R m1+k(α,q(α)) := (ψ k h,div v h ) Ωh (s κ), k = 1,...,m 2, where m 1 v h := v h (α) = v 0 + q k (α)ϕ k h, k=1 m 2 p h := p h (α) = q m1+k(α)ψh k k=1 and {ϕ k h }, {ψk h } is a basis of W 0h(s κ ) and L h (s κ ), respectively. The cost function J : R m R does not depend explicitely on α U since J does not. We recall the apriori estimates: v h 2 2,Ω h (s κ) + M D(v h) 3 3,Ω h (s κ) + div v h 3 3,Ω h (s κ) + v h2 3 3,Γ out + p h ,Ω h(s κ) C E, where C E > 0 is independent of h > 0 and s κ U κ ad. The following continuity property of the mapping α q(α), α U is a direct consequence of (T1) and (T2) (for the proof see [22]). 15

17 Lemma 9. Let α N α, N, where α N,α U, and let q(α N ) satisfy R(α N,q(α N )) = 0. Then there is a q(α) R m and a subsequence (denoted by the same symbol) such that and R(α,q(α)) = 0. q(α N ) q(α), N (31) Since U is compact, we immediately obtain the existence of a discrete optimal shape. Theorem 10. Problem (P n ) (and equivalently (P κh )) has a solution. 4.4 Convergence analysis The key role in our analysis plays the following counterpart of Theorem 3 (recall that the symbol stands for the zero extension of functions). Lemma 11. Let (s κ,v h (s κ ),p h (s κ )) G κh, h, κ 0+, s κ U κ ad, and α U ad satisfy s κ α in [0,L], κ 0 +. Then there exists v ( W 1,2 ( Ω) ) 2, p L 3 2 ( Ω) and appropriate subsequences such that ṽ h (s κ ) v in ( W 1,2 ( Ω) ) 2, M sκ D(ṽ h (s κ )) M α D( v) in ( L 3 ( Ω) ) 2 2, (32) p h (s κ ) p in L 3 2 ( Ω), h, κ 0 +. In addition, denoting v(α) := v Ω(α) and p(α) := p Ω(α), then v(α) Ŵv 0 (α) and (v(α),p(α)) solves ( P(α)). Remark 3. Since M α = 0 on Ω(α) \ Γ D, Mα is continuous in Ω. The same holds for the function l m,α. Proof of Lemma 11. We will proceed in the same way as in the proof of Theorem 15 in [3], with several minor changes. From (14) we know that the sequence { v h sκ, p h 3 2,Ω(sκ) } is bounded and that (32) holds for its appropriate subsequence. Since r h s κ α in [0,L] as h, κ 0+, we easily get that v(α) := v Ω(α) Ŵv 0 (α). In addition, p and v vanish in Ω \ Ω(α). From the density property of the system {L h } it follows that div v = 0 a.e. in Ω. We will focus on the limit passage in (P h (s κ )). Let ϕ V 0 (α) be given and ϕ h be the piecewise linear Lagrange interpolant of ϕ Ω(rh s κ) on the triangulation T h (s κ ) of Ω(r h s κ ). Since dist(supp ϕ,γ(r h s κ )) > 0 for h, κ > 0 small enough, the graph of r h s κ has an empty intersection with supp ϕ, which means that ϕ h W 0h (s κ ) and it can be used as a test function in (P h (s κ )). In addition, ϕ h ϕ in W 1, ( Ω) 2, h 0+, (33) 16

18 as follows from the well-known approximation results and the uniform regularity assumption (T3) on {T h (s κ )}. Now we can pass to the limit in the standard terms in (13): (D(ṽ h ), D( ϕ h )) bω (D( v), D( ϕ)) bω, ṽ h2 ṽ h2 ϕ h2 v 2 v 2 ϕ 2, (34) Γ out as follows from (32) and (33). Finally, in order to show that Γ out (ṽ hj ṽ hi x j, ϕ hi ) bω ( v j v i x j, ϕ i ) bω, ( p h,div ϕ h ) bω ( p,div ϕ) bω, h, κ 0+, ( div ṽ h div ṽ h,div ϕ h ) + Ãr h s κ (ṽ h ), ϕ h Ãα(ṽ(α)), ϕ, (35) we use the Vitali theorem. To prove pointwise convergence of D(ṽ h ) to D( v) we proceed as in the proof of Theorem 15 in [3]. Let Ω ε := {x Ω(α); dist(x, Ω) > ε}, ε > 0, and ξ := ξ ε C 0 (Ω(α)) such that ξ 0 in Ω(α) and ξ 1 in Ω ε. We construct a test function ϕ := ξ(ṽ h1 ṽ h2 ), where h 1,h 2 > 0 are sufficiently small so that ṽ h1 ṽ h2 3,supp ξ δ, ϕ 3,supp ξ δ (36) for arbitrary δ > 0. Instead of inserting ϕ directly into (P h1 (r h s κ )) and (P h2 (r h s κ )), we use the Lagrange interpolants ϕ h1, ϕ h2, respectively. We realize that if h i > 0, i = 1,2, is small enough, then We use (13) and (36) to deduce that ϕ hi ϕ 1,3,supp ξ δ. (37) 2µ 0 (D(v hi ), D(ϕ hi )) supp ξ + A sκi (v hi ),ϕ hi supp ξ = O(1), i = 1,2, (38) where O(1) denotes an expression which vanishes as δ 0. From the definition of ϕ, (36) and (37) we obtain: (D(v hi ),ξd(v h1 v h2 )) supp ξ = (D(v hi ), D(ϕ hi )) supp ξ + O(1), (39a) (M 3 s κ1 D(v hi )) D(v hi )),ξd(v h1 v h2 )) supp ξ i = 1, 2. Altogether, (37) (39) yield: = A sκi (v hi ), D(ϕ hi ) supp ξ + O(1), (39b) 2µ 0 D(v h1 v h2 ) 2 2,Ω ε 2µ 0 (D(v h1 v h2 ),ξd(v h1 v h2 )) supp ξ + 2ρ(ξM 3 s κ1 ( D(v h1 ) D(v h1 ) D(v h2 ) D(v h2 )), D(v h1 v h2 )) supp ξ Consequently = 2µ 0 (D(v h1 ), D(ϕ h1 )) supp ξ + A sκ1 (v h1 ),ϕ h1 supp ξ 2µ 0 (D(v h2 ), D(ϕ h2 )) supp ξ A sκ2 (v h2 ),ϕ h2 supp ξ = O(1). D(ṽ h ) D( v), h, κ 0+, a.e. in Ω for an appropriate subsequence. From this, (14) and the Vitali theorem we arrive at (35) (note that div v h = tr D(v h )). Thus (v(α),p(α)) solves ( P(α)). 17

19 Remark 4. As in the continuous case, due to the lack of a density result for W 0 (α), we are not able to prove that the limit v(α) belongs to W v0 (α). Therefore the augmented state problem ( P(α)) and shape optimization problem ( P) is considered instead of (P(α)) and (P), respectively. On the basis of the previous lemma we obtain the following convergence result. Theorem 12. Let v 0 3, b Ω be small enough so that the solutions of (P(α)) and ( P(α)), α U ad, are unique. Let {(s κ,v h,p h )} be a sequence of optimal pairs of (P κh ), h, κ 0+. Then there is a subsequence of {s κ,v h,p h )} such that s κ α in [0,L], ṽ h v in ( W 1,2 ( Ω) ) 2, M sκ D(ṽ h) M α D(v ) in ( L 3 ( Ω) ) 2 2, p h p in L 3 2 ( Ω), h, κ 0+, (40a) (40b) (40c) (40d) where (α,v Ω(α ),p Ω(α ) ) is an optimal triple for ( P). In addition, any accumulation point of {s κ,v h,p h )} in the sense of (40) possesses this property. Proof. Let α U ad be arbitrary. Then there exists a sequence {s κ }, s κ U κ ad, such that s κ α in [0,L], κ 0+, as follows from the well-known properties of Bézier functions. From Lemma 11 it follows that ṽ h (s κ ) v in ( W 1,2 ( Ω) ) 2, (41) M sκ D(ṽ h (s κ )) M α D(v) in ( L 3 ( Ω) ) 2 2, (42) p h (s κ ) p in L 3 2 ( Ω), h, κ 0+, (43) where (v h (s κ ),p h (s κ )) are the solutions of (P h (r h s κ )) and (v(α),p(α)) := (v Ω(α),p Ω(α) ) is the unique solution of ( P(α)). Since J is continuous with respect to convergence in (41) and we have that J(s κ,v h,p h) J(s κ,v h (s κ ),p h (s κ )), J(α,v Ω(α ),p Ω(α ) ) J(α,v(α),p(α)). Here α U ad is arbitrary, hence (α,v Ω(α ),p Ω(α )) is a solution of ( P). Remark 5. Let us mention that the state solutions must be unique for the complete convergence result. Otherwise the limit solutions are optimal only in a subclass of Ĝ formed by all accumulation points of solutions to (P h(r h s κ )), h, κ Numerical realization In this section we present a method of numerical realization of the shape optimization problem. We would like to emphasize that our implementation is not restricted to this particular problem but it can be applied to a wide range of shape optimization and optimal control problems that can be formulated like (P n ). 18

20 5.1 State problem We will start with the numerical solution of the discrete state problem (P h (α)) (see Section 3 for definition of (P h (α)), α Ũad) whose algebraic form is R(α,q(α)) = 0. (44) We assume that α is given and consider (44) as a system of m nonlinear algebraic equations for the vector of unknowns q := q(α) R m which will be solved by the Newton-Raphson method: ( ) 1 R Given q k R m, define q k+1 := q k q (α,q k) R(α,q k ). (45) Let us recall that the sequence {q k }, k = 0,1,..., converges provided that the initial guess q 0 is close enough to the solution of (44). Thus we have to supply a good approximation of q at the beginning. This is usually done by using some other algorithm (e.g. the fixed point iterations) before the Newton- Raphson method is used. The main advantage of this method is that if R is twice continuously differentiable and the inverse of R q (α,q(α)) exists, then convergence of (45) is at least quadratic. Instead of computing the inverse ( 1, R matrix q k)) (α,q we solve for every k the linear system R q (α,q k) q k = R(α,q k ) (46) for the unknown q k R m and put q k+1 := q k q k. For the solution of (46) we used the package SuperLU, which performs an LU decomposition with partial pivoting (see [6] for detailed description). Remark 6. It can be shown that the matrix R q (α,q k) for our problem is of the form ( ) R q (α,q Ak B k) = T k. B k O Moreover A k is positive definite provided that m 1 i=1 q2 ki is small enough. In our program we do not implement the analytical form of R q. Instead, we only specify how to assemble the residual vector R(α,q k ). The matrix of the linearized system (46) is obtained automatically by using tools of the automatic differentiation. The residual vector is decomposed into the sum of area and boundary integrals, which are further calculated element by element or edge by edge using suitable numerical quadratures. The algorithm for the numerical solution of the state problem now reads as follows: 1. Choose the tolerance r max > 0 for the residuum and the max. number of the Newton iterations k max N. 2. Choose q 0 R m. 3. For k = 0,...,k max 1: 19

21 Compute b k := R(α,q k ) and set C k := R(α,q k ) q. Solve C k q k = b k and set q k+1 := q k q k, r k := b k. If r k < r max then go to If r k < r max then set q := q k, otherwise report error. 5.2 Shape optimization problem We solve numerically the mathematical programming problem (P n ). Since the function to be minimized is smooth, we will use a gradient based minimization algorithm requiring certain gradient information. This is usually done by performing the sensitivity analysis, which can be quite involved in shape optimization (see [21]). In our approach we provide a simple algebraic sensitivity analysis of our cost function J. Most of the tedious work will be done by means of the automatic differentiation (see [12, 9]). For the numerical minimization itself we used the following packages: KNITRO - a robust tool for many types of smooth optimization problems (see [4, 5, 24]). NAG C library - in particular the function e04wdc which is intended for smooth optimization and uses the sequential quadratic programming [7]. Both packages provide a Fortran/C interface that allows to supply arbitrary routines for the cost function and gradient evaluation. A comparison of both packages and the obtained results can be found in Section 5.3 where results of several model examples are presented. The evaluation of the cost function J is done by the following chain: α q(α) J (α,q(α)). Here and in what follows we will consider more general case when J may depend explicitely on α. We assume also that the first mapping is single-valued, i.e. the state problem has a unique solution, so that J (α,q(α)) and J (α,q(α)) are well-defined. Hence we can define J(α) := J (α,q(α)). Note that the condition guaranteeing uniqueness of the solutions to the discrete state problems is independent of α and h. Now we describe how to compute the gradient J(α). The easiest way is to approximate it by difference quotients. However, this method is inexact, very sensitive on the choice of the step and time-consuming, especially when dealing with nonlinear state problems, since one gradient approximation requires (n+1) cost function evaluations. Instead we will implement the exact gradient evaluation using the adjoint equation technique which avoids differentiation of the control-to-state mapping: J α k (α) = J α k (α,q(α)) p ( R α k (α,q(α)) ), k = 1,...,n, (47) 20

22 where p := p(α,q(α)) is the solution of the adjoint equation: ( ) T R q (α,q(α)) p = J (α,q(α)). (48) q In this method we have to solve only one additional linear problem for p. Let us also notice that for our particular cost function used in the computations it holds that J α k (α,q) = 0, k = 1,...,n. However, the partial derivatives R α, R q, J q must be supplied to (47) and (48). Their hand-coding is in most cases elaborate and error-prone, requiring an additional algebraic sensitivity analysis. For this reason we compute them with the aid of the automatic differentiation. For the computations of the state problem for different α U (or s κ U κ ad, equivalently) we need to construct triangulations of Ω h (s κ ) that satisfy (T1) (T4). We use an approach which exploits the shape of Ω h (s κ ): We choose a suitable s κ U κ ad and create a triangulation T h(s κ ). Then, given s κ U κ ad, we define the triangulation of Ω h (s κ ) from T h (s κ ) in such a way, that every node (x(s κ ),y(s κ )) of T h (s κ ) is shifted in the vertical direction: x(s κ ) := x(s κ ), y(s κ ) := y(s κ ) s κ(x(s κ )) s κ (x(s κ )). (49) Due to the definition of Uad κ the assumptions (T1) (T4) are then valid. The evaluation of J and J can be summarized as follows: 1. Given α U, solve the state problem and obtain q(α); 2. Evaluate J(α) := J (α, q(α)); 3. Solve the adjoint equation (48) to obtain p(α); 4. Evaluate J using (47). 5.3 Model examples We end up with several numerical examples. Let us note that the parameters used in the following computations do not correspond to any real industrial application State problem Traditionally the paper machine header has been designed with a linearly tapered header. We use this header design to test the state problem solver. The computational domain is 9.5 m long and 1 m wide (see Figure 4). This domain is partitioned by using a uniform triangular mesh into 8000 triangles. The velocity is approximated by continuous piecewise quadratic functions, while the pressure by continuous linear functions. For this type of approximation (known as the Taylor-Hood element) it is known that the Babuška-Brezzi condition (11) holds true [2, 23]. The resulting number of degrees of freedom of q is

23 1 m 1 m 8 m 0.5 m 0.1 m Figure 4: Dimensions of the computational header. The pulp is modelled as an incompressible fluid with the laminar viscosity µ 0 = 10 3 Pa.s and the density ρ = 10 3 kg/m 3. The inlet and outlet velocity profiles are chosen as follows: v D {0} (0,H1) = (4(1 ( 2x 2 H 1 1) 8 ),0) m/s, v D {L} (0,H2) = (1 ( 2x 2 H 2 1) 8,0) m/s. If we define the kinematic viscosity ν := µ 0 /ρ, then ν 1 gives the Reynolds number 10 6 in case of the standard Navier-Stokes equations. This usually requires the use of a stabilized numerical scheme. However our turbulence model produces enough turbulent viscosity so that the state problem can be solved without any additional stabilization. As the initial approximation we chose a solution of a similar problem with a higher viscosity. The stopping criterion for the residuum is r max = The nonlinear loop needed from 2 to 10 iterations, each of which took about 7.1 s on AMD Opteron 246 with 2 GB RAM. The direct solver SuperLU was efficient enough for this problem size, requiring only 20% of one Newton iteration time, while the rest was spent on the residual assembly. Figure 5: Solution of the state problem (for σ = 10 3 and linearly tapered header): pressure p, velocity magnitude v and streamlines, dynamic viscosity µ Shape optimization problem The traditional linearly tapered header serves as a starting point for the shape optimization. The number of design parameters is set to n = 20. Due to the 22

24 1 0.1 KNITRO δ = 1 KNITRO δ = 0.1 NAG cost function e-04 1e iterations Figure 6: Convergence history of the used optimization algorithms. well-known properties of the Bézier functions the derivative of s κ Uad κ can be estimated as follows: s κ α max α min κ = α max α min n s κ U L ad. κ 2 Therefore for reasonably small n the constraint γ on the derivative will be removed from the definition of Uad κ. We then obtain a nonlinear optimization problem with simple bounds only. We set α max = H 1 and α min = H 2. The boundary segment Γ Γ out used in the definition of the cost function is Γ = (1.5,8.5). The outflow suction coefficient σ = 10 3 in what follows. We ran the computation repeatedly with two different target velocity profiles v opt. In the first case we used a constant target velocity v opt = m/s. We tested two optimization packages in order to compare the obtained results and the performance. All parameters were left default, only in case of KNI- TRO solver we tried several values of the initial trust region parameter δ. Both packages, KNITRO and NAG, converged apparently to the same shape. However NAG turned out to be superior in terms of the required cost function and gradient evaluations. KNITRO solver ended in all cases after approximately 100 iterations, achieving the KKT optimality conditions with the error smaller than On the other hand, NAG C library needed 73 iterations to get the optimality error smaller than The value of the cost function decreased from to in case of NAG. In Figure 6 the convergence history of all algorithms is shown. In the second case a function ( ) x L1 v opt = 0.65sin π m/s was chosen as the target outlet velocity. Here the computation ended after 44 iterations using NAG and the cost function value decreased from to L 2 23

25 Figure 7: Shapes of the header (from the top): Initial, optimal for the constant and non-constant target velocity Outlet velocity profiles initial optimal target Figure 8: Initial and optimal outlet velocity (constant target velocity). 24

26 Outlet velocity profiles initial optimal target Figure 9: Initial and optimal outlet velocity (non-constant target velocity) Computed optimal shapes are depicted in Figure 7. The optimal velocity profiles for the constant and the non-constant target are shown in Figure 8, and in Figure 9, respectively. There is no reason to expect that the cost function is convex, therefore the found minima are possibly only local ones. However, all the used algorithms converged to very similar shapes that are close to the one obtained in [12], where a different method was applied. Thus, there is a chance that the final design is close to the global minimum. In any case, for practical purposes it is usually sufficient to find a local minimum which improves the initial state. One can see from Figure 7 that the difference between the initial and optimized shapes is not too big. This indicates that the cost function is very sensitive with respect to shape variations. In spite of this fact, the proposed ex- Figure 10: Solution of the state problem: pressure p, velocity magnitude v and streamlines, dynamic viscosity µ, optimal shape for the constant target velocity. 25

27 Figure 11: Solution of the state problem: pressure p, velocity magnitude v and streamlines, dynamic viscosity µ, optimal shape for the non-constant target velocity. amples reveal that it is possible to control the outflow velocity and consequently the quality of produced paper by appropriate change of the header geometry. Conclusion The paper consists of 5 sections. After explaining the physical motivation in Section 1, we formulate the problem and recall the existence results for the continuous case in Section 2. Due to an algebraic turbulence model the weak formulation of the state problem involves the weighted Sobolev spaces. The main part of the paper is devoted to approximation and numerical realization of the problem formulated beforehand: In Section 3 a finite element discretization of the flow problem is studied. The existence of discrete solutions and their convergence to a solution of the continuous problem is proved. Section 4 describes an approximation of shapes, existence of discrete optimal shapes, and their convergence to a solution of the original shape optimization problem. The results of these two sections are obtained using the technique developped in [3], sharing many similarities with [15, 16, 17] and [19]. Finally, an algorithm for numerical realization is described in Section 5. The proposed method takes the advantage of the automatic differentiation which significantly simplifies and speeds up the computer program. The model examples show that very good results can be obtained and that the mathematical modelling together with numerical analysis can bring a significant contribution to the paper making engineering. Acknowledgement Jaroslav Haslinger acknowledges the research project MSM financed by MŠMT and A of the grant agency of the Czech Academy of Sciences (GAAV). Jan Stebel was supported by the Nečas Center for Mathematical Modelling project LC06052 financed by MŠMT. 26

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