Boundary control for non-isothermal Navier-Stokes flows by using domain decomposition and extended flow methods

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c

° TU Delft, The Netherlands, 2006

BOUNDARY CONTROL FOR NON-ISOTHERMAL

NAVIER-STOKES FLOWS BY USING DOMAIN

DECOMPOSITION AND EXTENDED FLOW METHODS

S. Manservisi

Nuclear Engineering Laboratory of Montecuccolino, Universit`a di Bologna, Via dei Colli 16, 40136 Bologna, Italy

and

Mathematics and Statistics Department,

Texas Tech University, Lubbock, Texas 79409-1042, USA e-mail: sandro.manservisi@mail.ing.unibo.it

Key words: Boundary Control, Navier-Stokes Equation, Energy Equation, Fluid Dy-namics

Abstract. In this work we study a class of stationary optimal temperature and optimal

flow control problems described by velocity, pressure and temperature where the optimal design of these systems is reached by controlling the boundary conditions. The optimal boundary control problem is transformed into an extended distributed problem and then solved by using standard distributed control techniques over the extended part of the do-main. The coupled multigrid solver for the optimality system is based on a local Vanka-type solver for the non-isothermal Navier-Stokes and adjoint system. The solution is achieved by solving and relaxing element by element the optimal control problem which is formulated by using an embedded domain approach. By using this technique only local subsystems are stored and solved which allows this method to be very effective in large problems where other solvers cannot satisfy memory and cpu requirements. Also the adjoint and the orig-inal system can be solved exactly over small domains leading to a very robust optimization. Some numerical examples of boundary controls are presented.

1 INTRODUCTION

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is stable and reliable. In this work we plan to applied these numerical techniques to the non-isothermal incompressible Navier-Stokes system. The optimal boundary control of the Navier-Stokes equations coupled with the energy equation has been considered by some authors, e.g., [8, 14, 12, 22, 23, 21]. The coupling between the state and adjoint equations in the nonlinear optimality system is strong and the solution should be found by using non-segregated methods which solve for all adjoint and state variables at the same time.

In this paper, we study a class of optimal flow control problems and its multigrid implementation for which the fluid motion is controlled by velocity forcing, i.e., injection or suction, along a portion of the boundary, and the cost or objective functional is a measure of the discrepancy between the temperature of the system and a given target temperature. We consider the two-dimensional incompressible flow of a viscous fluid on

Γ

Ω

Γ

Ω

1 2 3 4 5 6 8 1

Γ

₇

Γ

Ω

Γ

Ω

1 2 3 4 5 6 7 8 1 2

Figure 1: Flow domain Ω with the controlled boundary Γc= Γ7 (on the left) and extended flow domain

b

Ω = ∪Ω ∪ Ω2 (on the right).

the domain Ω with boundary Γ =S8

i=1Γi. The domain is shown in Fig.1 on the left with

boundary control over Γc = Γ7. The velocity ~u, the pressure p and the temperature T

satisfy the steady non-isothermal Navier-Stokes system

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∇ · ~u = 0 in Ω (1) (~u · ∇)T = α4T + Q in Ω

along with the Dirichlet boundary conditions

~u = ~g =      ~g1 on Γ1 ~gc on Γc= Γ5∪ Γ7 ~0 on Γ3∪ Γ4∪ Γ6∪ Γ8, (2) T = Tb =      T1 on Γ1 Tc on Γc= Γ5∪ Γ7 T6 on Γ6, (3) where ~f , Q are the given body force and the heat source respectively. If the boundary

condition is not specified in (2-3) we may assume homogeneous Neumann boundary condi-tions. In (1) ν and α denote the inverse of the Reynolds number Re and the inverse of the Prandl × Reynolds number whenever the variables are appropriately nondimensionalized. The vector ~g1 is the given velocity at the inflow Γ1 with temperature T1 and the vector ~gc

is the boundary control over Γc. Injection and suction can be performed through Γ7 with

fluid at temperature Tcand boundary velocity ~gc. Along the boundary Γ3∪ Γ4∪ Γ8 of the

adiabatic channel the velocity vanishes and along the surface Γ5, Γ6 the fluid exchanges

heat with the wall at temperature Tc and T6 respectively. The function ~g must satisfy the

compatibility condition _Z

Γ~g · ~n ds = 0 , (4)

where ~n is the unit normal vector along the surface Γ.

In the optimal control problem we would like to force the temperature over the do-main Ω ⊂ Ω to a desired distributione T by using velocity boundary control. There is ae

substantial literature discussing the set of all possible boundary controls (see for example [10, 17]). The function ~gc must belong to H1/2(Γc), the Sobolev space of order 1/2.

How-ever H1/2_(Γ

c) or H1(Γc) may not be sufficient to explicitly derive a first-order necessary

condition. Thus in general the set of all admissible controls ~g must be restricted to more regular spaces, namely, to belong to H3/2_(Γ

c). In order to satisfy this requirement the

standard steady optimal control problem is formulated in literature by using the following functional (see for example [12])

J = γ 2 Z e Ω|T − e T |2_{d~x +} β 2 Z Γc (|~g|2_{+ β} 1|~gx|2) d~xdt , (5)

where the minimization of the first term involving (T −T ) is the real goal of the temper-e

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the existence of the first order necessary condition for optimality through an appropriate choice of the positive coefficients β and β1 but the optimal control based on this

admis-sible set of solutions and the choice of β and β1 is not very friendly from the numerical

point of view and it turns out to be a very difficult task if injection or suction boundary velocity is required to satisfy the integral constraint (4).

In order to avoid these numerical problems we introduce an extended domain and transform the boundary control problem into a distributed control problem over the ex-tended domain. Specifically, we extend the model domain of Figure 1 along the line of control Γc. As in Figure 1 on the right we assume that all the controlled parts of Γc are

contained in the extended domain Ω.b

Now we reformulate the two-dimensional problem for a viscous incompressible flow over the region Ω = Ω ∪ Ωb 1∪ Ω2 by using distributed controls. The velocity u, the pressureb pb

and the temperature T satisfy the stationary systemb

−ν4u + (b u · ∇)b u + ∇b p = χb Ω1∪Ω2fb in Ωb (6)

∇ ·u = 0b in Ωb (7)

−α4T + (b u · ∇)b T = Qb in Ω .b (8)

The heat distributed source Q is set to zero. The function f is now the control andb χΩ1∪Ω2 is the characteristic function over Ω1∪ Ω2. The vectors ~g on Γc is the trace of ub and satisfies automatically the compatibility condition (4). We substitute the boundary control with the trace ofu which is to be determined by the associated distributed optimalb

control over the extended domain. The new cost functional becomes

J = γ 2 Z e Ω| b T −T |e 2d~x + β 2 Z Ω1∪Ω2 | ˆf |2d~x , (9)

and the new control problem is to find u,b p,bT ,b f and the trace ofb u over Γb c such that the

functional (9) is minimized subject to the system (6)–(8).

This new approach is numerically more friendly than the previous one. The resulting optimality system includes only the non-isothermal Navier-Stokes system and its adjoint. The vector ~g obeys to the compatibility condition (4) and normal controls may be included reducing the computational load.

The plan of the rest of the paper is as follows. In the next section, we introduce some notations and consider distributed optimal control associated to the boundary value problem for the extended domain. In the section 3 we show numerical experiments and its multigrid implementation for which the fluid motion is controlled by velocity forcing, i.e., injection or suction, along a portion of the boundary.

2 THE STATIONARY BOUNDARY CONTROL PROBLEM

We denote by Hs_{(O), s ∈ <, the standard Sobolev space of order s with respect to}

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(f, g)m and (f, g) denotes the inner product over H0(O) = L2(O). Hence, we associate

with Hm_{(O) its natural norm kf k} m,O =

q

(f, f )m. Whenever possible, we will neglect

the domain label in the norm. For vector-valued functions and spaces, we use boldface notation. For example, Hs_{(Ω) = [H}s_(Ω)]n _{denotes the space of <}n_{-valued functions such}

that each component belongs to Hs_{(Ω). Of special interest is the space}

H1(Ω) = ( vj ∈ L2(Ω) ¯ ¯ ¯ ∂vj ∂xk ∈ L2(Ω) for j, k = 1, 2 )

equipped with the norm k~vk1 = (

P₂

k=1kvkk21)1/2.

For Γs⊂ Γ with nonzero measure, we also consider the subspace

H1_Γ_s(Ω) = { ~v ∈ H1(Ω) | ~v = ~0 on Γs} .

Also, we write H1

0(Ω) = H1Γ(Ω). For any ~v ∈ H1(Ω), we write k∇~vk for the seminorm.

Let (H1 Γs)

∗ _{denote the dual space of H}1

Γs. Note that (H 1 Γs)

∗ _{is a subspace of H}−1_(Ω),

where the latter is the dual space of H1

0(Ω). The duality pairing between H−1(Ω) and

H1

0(Ω) is denoted by < ·, · >.

Let ~g be an element of H1/2_{(Γ). It is well known that H}1/2_{(Γ) is a Hilbert space with}

norm

k~gk1/2,Γ = inf ~v∈H1_{(Ω); γ}

Γ~v=~g

k~vk1,

where γΓ denotes the trace mapping γΓ : H1(Ω) → H1/2(Γ). We let (H1/2(Γ))∗ denote

the dual space of H1/2_{(Γ) and < ·, · >}

Γ denote the duality pairing between (H1/2(Γ))∗

and H1/2_{(Γ). Let Γ}

s be a smooth subset of Γ. Then, the trace mapping γΓs : H1(Ω) →

H1/2_(Γ

s) is well defined and H1/2(Γs) = γΓs(H1(Ω)).

Since the pressure is only determined up to an additive constant by the Navier-Stokes system with velocity boundary conditions, we define the space of square integrable func-tions having zero mean over Ω as

L2

0(Ω) = { p ∈ L2(Ω) |

Z

Ωp d~x = 0 } .

In order to define a weak form of the non-isothermal Navier-Stokes system, we introduce the continuous bilinear forms

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and the trilinear forms c( ~w; ~u, ~v) = Z Ω( ~w · ∇~u) · ~v d~x ∀ ~w, ~u, ~v ∈ H 1_{(Ω) .} ₍₁₃₎ d(~u; T, v) = Z Ω(~u · ∇T ) v d~x ∀ ~u ∈ H 1_{(Ω), T, v ∈ H}1_{(Ω) .} ₍₁₄₎

Obviously, a(·, ·) and k(·, ·) are continuous bilinear forms on H1_{(Ω) × H}1_{(Ω) and on} H1_{(Ω) × H}1_{(Ω) respectively. b(·, ·) is a continuous bilinear form on H}1_{(Ω) × L}2

0(Ω)

and c(·; ·, ·), d(·; ·, ·) are a continuous trilinear form on H1_{(Ω) × H}1_{(Ω) × H}1_{(Ω) and}

H1_{(Ω) × H}1_{(Ω) × H}1_{(Ω) respectively. For details concerning the function spaces we have}

introduced, one may consult [2, 27] and for details about the bilinear and trilinear forms and their properties, one may consult [11, 27].

We now formulate the mathematical model of the optimal boundary control problem. Let Ω be an extended domain andb Γ be the corresponding boundary. If Γb c is the part of

the boundary where we apply the control we assume that Γ − Γc is a subset of Γ, namelyb

only the controlled part of the boundary lies inside the extended domainΩ. Letb Γbdbe the

part of Γ where the velocity Dirichlet boundary conditions are applied. Overb Γbe=Γ \b Γbd

only homogeneous Neumann boundary conditions for the velocity field are considered. We define in a similar wayΓbf and Γbg for the Dirichlet and Neumann boundary conditions

for the temperature T over Γ respectively. In the rest of the paper we denote by u theb restriction to Ω of a function u defined over the domainb Ω and vice-versa.b

The optimal boundary control problem can then be stated by using the extended domainΩ and the distributed extended forceb f in the following way:b

find f ∈ Lb 2_(Ω

1∪ Ω2) such that (u,b p,b T ,b τ ) minimizes the functionalb J = γ 2 Z e Ω| b T −T |e 2_{d~x +} β 2 Z Ω1∪Ω2 | ˆf |2_{d~x ,} ₍₁₅₎ and satisfies                            a(u,b v) + c(b u;b u,b v)+ <b τ ,b v >b _Γˆ +b(v, p) =<b f ,b v >b ∀v ∈ Hb 1(Ω)b b(u,b q) = 0 ∀b q ∈ Lb 2 0(Ω)b k(T ,b r) + d(b u;b T ,b r) = 0 ∀b r ∈ Hb 1(Ω)b <u,b s >b _b_Γ d= <g,b s >b bΓd ∀bs ∈ H −1/2₍_Γb d) <T ,b _{t >}b b Γf = < c Tb,bt >_b_Γ_f ∀t ∈ Hb −1/2(Γbf) (16)

with f = ~b f over Ω. The domain Ω is the part of the domain Ω over which the matchinge is desired. The corresponding boundary control ~gc can be found after the solution of

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note that the boundary control ~gc automatically satisfies the compatibility condition (4).

Note that solutions of (16) exists for any value of the Reynolds number. However the uniqueness can be guaranteed only for “large enough” values of ν or for “small enough” data ~f and ~g. The admissible set of states and controls is given by

Aad = {(~u, p, T,f , ~gb c) ∈ H1(Ω) × L20(Ω) × H1(Ω) × L20(Ω) × Hb 3/2(Γc) with ~gc= γΓcu andb f = ~b f over Ω

such that J (T ,b f ) < ∞ and (b u,b p,b T ) satisfies (16)} .b

The existence of optimal solutions in this admissible set can be studied by using standard techniques (see for example [1, 10, 15, 16, 17, 28]). Following this approach it is possible to show that optimal control solutions must satisfy a first-order necessary condition. They must satisfy the following system of equations

                           νa(u,b v) + c(b u;b u,b v) + b(b v,b p) =<b f ,b v >b ∀v ∈ Hb 1 b Γd( b Ω) b(u,b q) = 0 ∀b q ∈ Lb 2 0(Ω)b k(T ,b r) + d(b u;b T ,b r) = 0. ∀b r ∈ Hb 1 bΓf( b Ω) <u,b s >b _b_Γ d=<g,b s >b bΓd ∀s ∈ Hb −1/2₍_Γb d) <T ,b bt >_b_Γ f=< b Tb,bt >_b_Γ_f ∀bt ∈ H−1/2(Γbf) , (17)

and the adjoint system

                             νa(w,b v) + c(b w;b u,b v) + c(b u;b w,b bv) + b(v,b σ) + d(b v;b T ,b R) = 0 ∀b v ∈ Hb 1 b Γd( b Ω) b(w,b q) = 0 ∀b q ∈ Lb 2 0(Ω)b k(R,b r) + d(b u;b r,b R) + γb Z e Ω( b T −T )e r d~x = 0. ∀b v ∈ Hb 1 b Γf( b Ω) <w,b s >b _b_Γ d= 0 ∀s ∈ Hb −1/2₍_Γb d) , <R,b t >b _b_Γ f= 0 ∀ b t ∈ H−1/2(Γbf) , (18) with ~gc= γΓcub (19)

and f = ~b f over Ω and f =b w/β over Ωb 1∪ Ω2. The optimality system for the boundary

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the compatibility constraint is a limit to the feasibility of the normal boundary control. The normal boundary control must obey to this integral constraint reducing enormously the possibility to achieve accurate and fast numerical solutions of the necessary optimal control system with non-embedded techniques.

3 NUMERICAL IMPLEMENTATION OF THE BOUNDARY CONTROL

PROBLEM

The optimal boundary control problem can be solved by using a multigrid approach and the multigrid smoothing operator for each grid level can be derived directly from the optimal control problem. There is a vast class of smoothing operators for multigrid methods but we are interested in the class of Vanka-type solvers. In this class of solvers, which are well known for solving Navier-Stokes equations, the iterative solution is achieved by solving several exact systems involving blocks of variables. In particular we use the close relationship between this class of solvers and the class of solvers arising from saddle point or minimization problems which allows us to use conforming standard finite elements.

(a) (b)

(d) (c)

Figure 2: Domain bΩh (a) with the mesh at the level 0 (b), level 1(b) and level 2(c).

LetΩbhbe the square geometry described in Fig.2 (a). Now, by starting at the multigrid

coarse level l0 we subdivide Ωbh into triangles or rectangles by families of elements Thi,l0.

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to mesh the entire domain Ωbh at the top finest multigrid level lnt. For example in Fig.2

(b) the mesh at the grid level l1 is obtained by simple midpoint refinement from the mesh

in Fig.2 (a) at the level l0. With successive refinements we obtain the mesh in (d) at the

level l2.

At the multigrid level l we introduce the approximation spaces Xhl ⊂ H 1₍_Ωb

h), Shl ⊂

L2₍_Ωb

h) and Xhl ⊂ H1(Ωbh) for the velocity, pressure and temperature respectively . The

approximate function obeys to the standard approximation properties including the LBB-condition. Let Phl = Xhl|_∂b_Ω, i.e., Phl consists of all the restrictions, to the boundary ∂Ω,b

of functions belonging to Xhl. For all choices of conforming finite element space Xh we

then have that Phl ⊂ H −1

2(∂Ω). See [7, 11] for details concerning these approximationb spaces. The extended velocity, pressure and temperature fields (ubhl,pbhl,Tbhl) ∈ Xhl(Ωbh) ×

Shl(Ωbh) × Xhl(Ωbh) at the level l satisfy the system of equations

                                   a(ubhl,vbhl) + c(ubhl;ubhl,vbhl) + b(vbhl,pbhl) =<fbhl,vbhl > ∀vbhl ∈ Xhl(Ωbh) ∩ H 1 b Γdh( b Ωh) b(ubhl,rbhl) = 0 ∀rbhl ∈ Shl(Ωbh) k(Tbhl,rbhl) + d(ubhl;Tbhl,rbhl) = 0 vbhl ∈ Xhl(Ωbh) ∩ H 1 b Γf h( b Ωh) <ubhl,sbhl >b_Γ_dh= < ~g,sbhl >b_Γ_dh ∀sbhl ∈ Phl(Γbdh) <Tbhl,tbhl >b_Γ_{f h}= < ~Tb,tbhl >b_Γ_{f h} ∀bthl ∈ Phl(Γbf h) (20)

and the adjoint

                                     a(wbhl,vbhl) + c(wbhl;ubhl,vbhl) + c(ubhl;wbhl,vbhl) + b(vbhl,σbhl) + d(vbhl;Tbhl,Tbhl) = 0 ∀vbhl ∈ Xhl(Ωbh) ∩ H 1 bΓh−Γ2h( b Ωh) b(wbhl,qbhl) = 0 ∀qbhl ∈ Shl(Ωbh) k(Rbhl,bthl) + d(ubhl;tbhl,Rbhl) + γ Z e Ω( b Thl−T ) ·e tbhld~x = 0 ∀bthl ∈ Xhl(Ωbh) <wbhl,sbhl >b_Γ_dh= 0 ∀sbhl ∈ Phl(Γbdh) <Rbhl,bthl >b_Γ_{f h}= 0 ∀tbhl ∈ Phl(Γbf h) (21) with ~gchl = γΓcubhl (22)

and fbhl = ~fh over Ωhl and fbhl =wbhl/β over Ωbhl− Ωhl.

The unique representations of ubhl,wbhl, Tbhl, Rbhl and pbhl, σbhl as a function of the nodal

point values ubl(k1), wbl(k1), Tbl(k1), Rbl(k1) and pbl(k2), σbl(k2) ( k1 = 1, 2, ...nvt with nvt =

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points) define the finite element isomorphisms Φl : Ul → Xhl, Φ+l : Wl → Xhl, χl : Vl → Xhl, χ+l : Zl → Xhl Ψl : Πl → Shl Ψ+l : Σl → Shl between the vector spaces Ul, Wl,Vl, Zl, Πl, Σl of nvt-dimension and npt-dimension vectors and the finite element spaces Xhl,

Xhl, Shl.

At the level l we introduce the corresponding finite element matrices Al, Bl, Cl(ubhl),

Kl, Dl(ubhl) for the discrete Navier-Stokes operators a, b, c, k , d defined by (10-13)

respectively. Their corresponding finite element matrices for the adjoint operators are denoted by A+

l , Bl+, Cl+(ubhl), K +

l , D+l (ubhl). The Navier-Stokes/adjoint coupled terms

are denoted by Hl and Gl(Rbhl). Now the problem (20-21) is equivalent to

          Al+ Cl BlT 0 Hl/β 0 0 Bl 0 0 0 0 0 0 0 Kl+ Dl 0 0 0 0 0 Gl A+l + Cl+ (B+l )T 0 0 0 0 B+ l 0 0 0 0 0 0 0 K_l++ D+_l + γIl                     b uhl,n b phl,n b Thl,n b whl,n b σhl,n b Rhl,n           =            b Fhl 0 b Lhl 0 0 c Mhl           

at the multigrid level l. In the vector spaces Ul, Wl,Vl,Zl, Πl and Σl we use the usual

Euclidean norms which can be proved equivalent to the norms introduced to the corre-sponding finite element approximation spaces (see [6, 19] for details).

Essential elements of a multigrid algorithm are the velocity, temperature and pressure prolongation maps

Pl,l−1(u) : Ul−1 → Ul Pl,l−1(T ) : Vl−1 → Zl Pl,l−1(p) : Πl−1 → Πl (23)

and the velocity and restriction operators

Rl−1,l(u) = P∗l,l−1(u) : Ul→ Ul−1 Rl−1,l(T ) = Pl,l−1∗ (T ) : Vl→ Zl−1 (24) Rl−1,l(p) = Pl,l−1∗ (p) : Πl → Πl−1.

Since we would like to use conforming Taylor-Hood finite element approximation spaces we have the nested finite element hierarchies Xh0 ⊂ Xh1 ⊂ ... ⊂ Xhl and Sh0 ⊂ Sh1 ⊂ ... ⊂

Shl and the canonical prolongation maps Pl,l−1(u), Pl,l−1(p), Pl,l−1(u) can be obtained

simply by

Pl,l−1(u) = Φl−1(Φ−1l (u))

Pl,l−1(p) = Ψl−1(Ψ−1l (p)) . (25) Pl,l−1(T ) = Ψl−1(Ψ−1l (p)) .

For details and properties one can consult [19, 25] and citations therein.

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Navier-Stokes equations (see [20, 29]). An iterative coupled solution of the linearized and discretized incompressible Navier-Stokes equations requires the approximate solution of sparse saddle point problems. In this multigrid approach the most suitable class of solvers is the Vanka-type smoothers. They can be considered as block Gauss-Seidel meth-ods where one block consists of a small number of degrees of freedom (for details see [29, 19, 20]). The characteristic feature of this type of smoother is that in each smoothing step a large number of small linear systems of equations has to be solved. In the Vanka-type smoother, a block consists of all degrees of freedom which are connected to few neighboring elements. As shown in Fig.3 for conforming finite elements the block could

a b

Figure 3: Blocks of unknowns: 21V + 1P (on the left) and 16V + 4P (on the right)

consist of all the elements containing a pressure vertex or four pressure nodes, namely 21 velocity nodes (circles and squares) with one pressure node (square) or 16 velocity nodes (circles and squares) with four pressure nodes (squares) respectively. Thus, in the first case a relaxation step with this Vanka-type smoother consists of the iterative solution of the corresponding block of equations over all the pressure nodes. In the second case a re-laxation step consists of the solution of the block of equations over all the elements where the velocity and pressure variables are updated iteratively. Different blocks of unknowns can be solved including local constraints as they arise from the optimal control problem. For convergence and properties of this class of smoothers one can consult [29, 19, 20] and citations therein.

4 BOUNDARY CONTROL TEST

In this numerical experiment we would like to illustrate an example where boundary controls can be efficiently applied to real situations. We consider a channel where the inflow over Γ1 is assigned and the temperature near to the output Γ2 must be controlled

by injection or suction along a portion of the boundary Γc.

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Γ

Ω

Γ

Ω

1 2 3 4 6 8 1

Γ

₇ 50 C

Γ

₅ 300 C 150 C 150 C

Γ

Ω

Γ

Ω

2 3 4 5 6 7 8 1 2

Ω

Γ

1

Figure 4: The domain (on the right) and the extended domain bΩ at grid level l0 (on the left) with the

controlled subdomain eΩ. 0 0.2 0.4 0.6 0.8 1 x 1.6 1.65 1.7 1.75 1.8 1.85 T/100 0 0.2 0.4 0.6 0.8 1 x 1.99 1.995 2 2.005 2.01 T/100 0 0.2 0.4 0.6 0.8 1 x 2.19 2.195 2.2 2.205 2.21 T/100

Figure 5: Temperature profile (solid) over Γ2and the corresponding desired profile (dashed) for the case

A, B and C from left to right respectively

design and represent the area where the fluid may be controlled. If a control is active in that area then we model such a control as a boundary control, remove the cavity from the domain and use that cavity as a part of the extended domain Ω as shown in Figure 4b on the left. In this paper we study the case in which only Γ7 is controlled and the desired

temperature field T is constant over the controlled areae Ω. The inflow boundary Γe 1 has

a parabolic velocity profile (max velocity 1m/s) at the temperature T = 50o_{C. Over}

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0 0.5 1 1.5 2 2.5 3 y 0 0.5 1 1.5 2 2.5 3 T/100 A B C

Figure 6: Temperature profiles along the centerline for different temperature targets T = 175o_{C (C),}

200o_{C (B) and 220}o_{C (A).} 0 0.5 1 1.5 2 2.5 3 y 0 0.5 1 1.5 2 2.5 3 v C B A

Figure 7: Velocity profiles along the centerline for different temperature targets T = 175o_{C (C), 200}o_C

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0.25 0.5 0.75 y -2 -1 0 1 2 u C B A 0.25 0.5 0.75 y -0.2 -0.1 0 0.1 0.2 0.3 v A B C

Figure 8: Control components u and v over Γ6 (bottom) for different temperature targets T = 175oC

(C), 200o_{C (B) and 220}o_{C (A).}

is kept at 150o_{C. The surface Γ}

6 exchanges heat with the fluid at temperature 300oC

and over all the rest of the boundary homogeneous Neumann boundary conditions are considered for the temperature field. Outflow boundary conditions are considered over Γ2 with no-slip boundary conditions over the cannel walls. In all these computations we

assume µ = 0.01 and α = 0.25. The extended domain Ω = Ωb 1 ∪ Ω2 is shown in Fig.1

and in Fig.4 on the right. The mesh levels l1,l2, l3, l4 and l4 over the extended domain Ωb

are generated by midpoint refinement starting from the mesh at the level l0 as shown in

Fig.4. The boundary conditions are imposed on this coarse mesh and then imposed over the other levels by using the standard multigrid interpolation operator.

In order to evaluate the ability of the method to control the outflow over Γ2 we set

different temperatures in Ω and compute the corresponding control over Γe 7. This system

is designed to mix the flow at temperature T1 = 50oC given by the inflow boundary

condition with the controlled flow over Γ7 at Tc= 150oC. Since the main source of heat

is the surface Γ6 the control should increase or decrease the flow rate through the channel

Ω in order to modify the outflow temperature. The increasing of the channel flow should decrease the outflow temperature and viceversa. We label A, B and C the computations when the desired outflow temperature over Ω is 220e o_{C, 200}o_{C and 175}o_{C respectively.}

In the Figures 5-8 the results for different values of the target temperatures are shown. Since the first term of the functional is rather large in all these computations we set

γ = 10−6 _{and β = 10}−4_{. In Figure 5 we note that the average temperature profile is very}

well matched for all the temperature targets. However the profile cannot be matched along all the section since the control boundary Γ7 is rather far from the controlled areaΩ. Thee

temperature and the velocity solutions along the channel centerline and the control over Γ6 is reported in Figure 6 and Figure 7 respectively.

In Figure 6 we have the temperature profiles along the centerline with the corresponding desired temperatures (dashed lines) for T = 175o_{C (C), 200}o_{C (B) and 220}o_{C (A). We}

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0 0.2 0.4 0.6 0.8 1 x 1.6 1.7 1.8 1.9 2 T/100 A B C D

Figure 9: Desired (D) and computed temperature profiles along Γ2for different β = 5×10−3(A), 5×10−4

(B) and 5 × 10−5 _{(C) (γ = 1.10}−6₎ 0 0.5 1 1.5 2 2.5 3 y 1 1.5 2 2.5 3 v A B C 0 0.5 1 1.5 2 2.5 3 y 0.6 0.8 1 1.2 1.4 1.6 1.8 2 T/100 A B C

Figure 10: Temperature and velocity profiles along the centerline for different β = 5 × 10−3_{(A), 5 × 10}−4

(B) and 5 × 10−5 _{(C) (γ = 1.10}−6₎

good over the matching domainΩ. In order to control the outlet temperature the channele flow must increase from case A to C and so the corresponding velocity along the centerline as shown in Fig.7. We note that for low temperature targets the inflow imposed over Γ1

is not sufficient to cool down the surface Γ6 and the control must increase the flow. At

the contrary for high temperature target the inflow over Γ1 is sufficient to cool down the

surface Γ6 and the control must drain fluid from the main branch of the channel through

the control boundary Γ7. This can easily seen on the horizontal velocity component u

shown on the bottom of Fig.8. We note that the controlled normal component of the boundary control may be positive and negative, namely there is injection (case C) and suction (case A and B).

In Figures 9-11 we show the solution for decreasing values of the penalty parameter β for the case C previously described. In these figures β is equal to 5 × 10−3 _{(A),5 × 10}−4

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0.25 0.5 0.75 y -2 -1.5 -1 -0.5 0 u A B C 0.25 0.5 0.75 y 0 0.1 0.2 0.3 0.4 v A B C

Figure 11: Control velocity components u and v over Γ6for different β = 5 × 10−3 (A),5 × 10−4 (B) and

5 × 10−5 _{(C) (γ = 1.10}−6₎

optimal solution. In Figure 9 we have the temperature outflow profile for different β and the desired temperature target (D). The boundary control is very effective to increase the flow with suction through the boundary Γc and to decrease the outflow temperature

as we can see in Figure 10. For different value of β ( β = 1 × 10−4 _{(A), β = 1 × 10}−3 _(B) β = 1 × 10−2 _{(C)) we can see the profile of the velocity component v and the temperature}

along the center line. Figure 11 shows the boundary control on Γc for β = 5 × 10−3 (A),

5×10−4 _{(B) and 5×10}−5_{(C). The u-component is shown on the left and the v-component}

on the right.

5 CONCLUSIONS

We have introduced an extended method for boundary controls which allows matching temperature field very efficiently. It is accurate and avoids the cumber-stone coupling of the boundary equation with the Navier-Stokes, energy and the adjoint system. This method allows to solve the problem for boundary controls which must obey to the com-patibility condition and boundary control corners. A particular class of multigrid solvers, which is a domain decomposition method at element level, is used in this paper to solve ex-actly the optimal control problem producing accurate and robust solutions. All this leads to improved computability and reliability for the numerical solution of steady boundary control.

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