SHAPE OPTIMIZATION WITH STOKES CONSTRAINTS OVER THE SET OF AXISYMMETRIC DOMAINS∗
MAITINE BERGOUNIOUX† ANDYANNICK PRIVAT‡
Abstract. In this paper, we are interested in the study of a shape optimization problem with Stokes constraints within the class of axisymmetric domains represented by the graph of a function.
Existence results with weak assumptions on the regularity of the graph are provided. We strongly use these assumptions to get some topological properties. We formulate the (shape) optimization problem using different constraints formulations: uniform bound constraints on the function and its derivative and/or volume (global) constraint. Writing the first order optimality conditions allows us to provide quasi-explicit solutions in some particular cases and to give some hints for the treatment of the generic problem. Furthermore, we extend the (negative) result of [A. Henrot and Y. Privat, Arch. Ration. Mech. Anal., 196 (2010), pp. 281–302] dealing with the nonoptimality of the cylinder.
Key words. shape optimization, dissipated energy, Stokes system, first order optimality condi- tions
AMS subject classifications.49Q10, 49J20, 49K20, 35Q30, 76D05, 76D55 DOI.10.1137/100818133
1. Introduction. The applications of shape optimization to fluid mechanics are numerous. Some well-known and studied situations may be encountered in industry, for instance, airplane optimization, where the drag is often minimized under a lift constraint (see, e.g., [7, 2, 24, 28] for examples of such applied studies). More gen- erally, the study of shape optimization problems in the context of fluid mechanics constitutes a challenge. Most of time, works on this topic are numerical point of view studies, because of the intrinsic difficulty of Stokes or Navier–Stokes equations.
Among the well-known and studied problems of shape optimization for fluids, one can mention, for instance, the reduction of the drag of an airplane wing in order to ensure hydrodynamic stability or the minimization of the noise of vortex shedding for designing the shape of an airfoil trailing edge.
The partial differential equation describing the behavior of the fluid appears then as an additional constraint for the optimization problem. For early references on this topic, we mention [14, 16, 28, 32, 33]. We also mention some closely related works.
In [37], the authors minimize the absolute value of the drag with respect to the shape of a particle in a unbounded quiescent fluid with constant velocity along an axis. As in the present paper, the behavior of the fluid is described by a Stokes system, and the shape of the particle is assumed to be axisymmetric. The main difference with our study lies in the fact that the domain of the fluid surrounding the particle is unbounded, and the authors chose to represent the solution of the Stokes system with an integral formulation. Such an approach would clearly not be adapted to our case, where the fluid is assumed to lie inside the solid, whose shape is optimized.
∗Received by the editors December 14, 2010; accepted for publication (in revised form) October 31, 2012; published electronically February 12, 2013.
http://www.siam.org/journals/sicon/51-1/81813.html
†Labo. MAPMO, CNRS, University d’Orl´eans, UMR 6628, F´ed´eration Denis Poisson, FR 2964, Bat. Math., BP 6759, 45067 Orl´eans cedex 2, France (maitine.bergounioux@univ-orleans.fr).
‡ENS Cachan Bretagne, CNRS, University Rennes 1, IRMAR, av. Robert Schuman, F-35170 Bruz, France (yannick.privat@bretagne.ens-cachan.fr). This author was partially supported by ANR Project OPTIFORM.
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In [3, 31], the authors study a Stokes problem, both from the theoretical and numerical points of view, in the same spirit as ours. They state optimality conditions, assuming the existence of an optimal shape.
The authors of [17, 18] led a theoretical study on the shape minimization of the dissipated energy. In particular, the first order optimality conditions for the optimization problem were written and exploited to prove that, under some given particular boundary conditions chosen to model the trachea in humans, the cylinder is not an optimal solution. Nevertheless, some numerical computations done in the same papers lead us think that the optimum is very close to a cylinder.
This paper is motivated by a simple question raised by the conclusions of [17], where it was proven that the cylinder does not optimize the dissipated energy through a pipe when the fluid inside is driven by Stokes or Navier–Stokes laws. Moreover, the problem of knowing whether or not the optimal solution has a cylindrical symmetry is pointed out and still open. Another point of view consists of imposing the cylindri- cal symmetry in the class of admissible shapes for this optimization problem. Such a choice can be justified by the fact that, in some situations, it is natural to make this assumption. For instance, if we assume that the shape of the (human) trachea minimizes the energy dissipated by the air through the geometry (thanks to a natural selection process), it is reasonable to consider only simply connected domains with cylindrical symmetry. With such a restriction on admissible shapes, we may hope for simplifications of the system driving the behavior of the fluid to obtain existence of optimal shape results more easily and a simple expression of the first order opti- mality conditions. In this paper, we decided to make the (strong) hypothesis that a transversal slice of an admissible domain is the graph of a function z →a(z). Our goal is to investigate the question of the existence of optimal shapes over the class of such cylindrical domains, with Stokes partial differential equations constraints, and write the first order optimality conditions. This study may be seen as preliminary, especially in view of a refined study of the qualitative properties of the optimum (for instance, the very difficult question of the free boundary regularity) and numerical computations.
As will be emphasized in the following sections, it is quite easy to prove the existence of an optimal shape for three dimensional domains and to ensure a strong convergence of the terms of the minimizing sequence of domains to the optimum.
Nevertheless, the theoretical and numerical applications previously mentioned need a precise framework adapted to the domains having a cylindrical symmetry. One of the difficult parts of our work lies in the determination of a variational formulation taking into account the cylindrical character of the domain and the symmetry properties of the solution of Stokes partial differential equations.
The paper is organized as follows. Section 2 is devoted to optimal shape existence.
We first give a generic abstract result, and then a specific result in the “axisymmetric graph” case. We use the cylindrical symmetry assumption to give a two dimensional formulation of the Stokes equation, using cylindrical coordinates. We use a fictitious domain technique to get results without strong regularity assumptions on the shape boundary. Optimality conditions are investigated in section 3. If no volume con- straint is added, we prove a generic monotonicity result of the cost functional for the inclusion of domains. As a result, the problem becomes purely geometrical, which allows us to provide a precise characterization of the optimum. The same shape opti- mization problem with an additional volume constraint appears rather difficult. We are nevertheless in position to establish the nonoptimality of the cylinder in that case
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(extending in the same way the results of [18]). Moreover, we propose some hints for writing the first order optimality conditions and preparing a future numerical work on that topic. Finally, proofs are postponed to Appendix A.
2. Some shape existence results.
2.1. Preliminaries. Let L, a0, and a1 be three strictly positive real numbers such thata0< a1. Let us introduce the set of admissible parametrizations
(2.1) A∞=
a∈W1,∞(0, L)|a0≤a(z)≤a1forz∈(0, L) .
We consider a generic domain Ωa, assumed for the moment to be simply connected, bounded, with Lipschitz boundary, and axisymmetric with respect to the (Oz)-axis.
More precisely, the domain expressed in standard cylindrical coordinates is (2.2) Ωa={(r, θ, z)∈R+×T×R+|0< r < a(z),0< z < L},
where Tdenotes the torus R/2πand a∈ A∞. We denote byDa the bounded open set, whose closure is
(2.3) Da={(z, r)∈R+×R+|0≤z≤L,0≤r≤a(z)},
so that Ωa is obtained by a rotation ofDa around the axis{r=θ= 0}denoted from now on as (Oz). We write
∂Ωa=Ea∪Σa∪Sa,
where Ea = ΩA∪ {z = 0} is the inlet surface, Sa = ΩA∩ {z = L} is the outflow surface, and Σa =∂Ωa\(Ea ∪Sa) is the lateral surface. Similarly, we write ∂Da = Γ0∪ΓL∪Γb∪Γa, where Γ0 ={(0, r)|0 ≤r≤a(0)}, ΓL={(L, r)|0≤r≤a(L)}, Γb = {(z,0) | 0 ≤ z ≤ L}, Γa = {(z, a(z))| 0 ≤ z ≤ L}. The notations used are summarized in Figure 2.1.
Fig. 2.1.A cylindrical admissible domain and its slice in two dimensions.
The same model of fluid as in [17, 18] is studied, for instance, to model the flow of the air inside the trachea represented by Da. More precisely, denoting by u the velocity of the fluid, withpits pressure, the Stokes system is written as
(2.4)
⎧⎪
⎪⎪
⎪⎨
⎪⎪
⎪⎪
⎩
−μΔu+∇p= 0 in Ωa,
∇ ·u= 0 in Ωa, u= 0 on Σa, u=u0 onEa, σ(u, p)·n=h onSa,
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where μ > 0 stands for the viscosity of the fluid, u0 is a Dirichlet datum, h is a Neumann-like datum to be made precise later, and where the standard tensorial notations of fluid mechanics are used; i.e.,
σ(u, p) =−pI3+ 2με(u) is the stress tensor of (u, p), and
ε(u) = 1 2
∇u+ (∇u) = 1
2 ∂ui
∂xj
+∂uj
∂xi
1≤i,j≤3
is the strain tensor ofu(symmetric part of the gradient tensor). We also define the trace inner product of two vector fieldsAandBofRd,d= 2,3, by
A:B= d i,j=1
AijBij.
Throughout the paper, bold letters stand for vector fields ofR2 orR3.
The existence of solutions for system (2.4) is well known (see, for instance, [4, 15]).
Theorem 2.1. Let a∈ A∞. Assume that u0∈(H3/2(Ea))3,h∈(H1/2(Sa))3; then problem(2.4)has a unique solution (u, p)∈(H1(Ωa))3×L2(Ωa).
Since we are dealing with Stokes equations, we have to assume that a(roughly speaking, the boundary of the domain) is smooth enough (say, for instance,a∈ A∞) to get regular solutions. The existence and regularity of solutions to Stokes equations in nonsmooth domains have not been thoroughly investigated. Let us nevertheless mention the works [8, 9, 26, 27, 35].
The assumption thata∈ A∞will be relaxed in what follows using a weak varia- tional formulation to define solutions of (2.4).
For the inlet boundary condition, we will chooseu0= (0,0, u0(r)), whereu0is a positive function of the polar variabler=
x21+x22.
For the outlet boundary condition, we will choose h= −p0n, where ndenotes the outward-pointing normal vector, andp0>0 is a real number. In particular, this boundary condition can model the human bronchial tree (see, e.g., [25]). With such a choice, introducing ¯p=p−p0, it is easy to see that the pair (u,p¯) is a solution of system (2.4), wherehhas been replaced by 0 in the boundary condition onSa. For this reason, we will actually chooseh= 0 in what follows.
Then, it is relevant to wonder whether we are able to rewrite the Stokes problem (2.4) using only cylindrical coordinates (r, z), as we can easily have this intuition, since the geometry is cylindrically symmetric. Moreover, the functional we want to minimize is the energy dissipated by the fluid (or viscosity energy) defined by
(2.5) J0(Ω) = 2μ
Ω
|ε(u)|2dx,
where u is the solution of system (2.4), in a certain class of admissible shapes parametrized, for instance, by the elements ofA∞.
We end this subsection with a standard, but essential, ingredient for the coming existence study. The solution of the Stokes problem (2.4) can be seen as the unique minimizer of an energy functional j. We recall this fact and its proof for the sake of completeness.
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Lemma 2.2. uΩa is the (unique) solution of(2.4)if and only ifuΩa is a solution of
(2.6)
⎧⎪
⎪⎨
⎪⎪
⎩
minj(u) = 2μ
Ωa
|ε(u)|2dx,
u∈Hdiv(Ωa) ={v∈[H1(Ωa)]3| ∇ ·v= 0in Ωa, v= 0on Σandv= (0,0,u0)on E}.
Proof. Let (un)n∈Nbe a minimizing sequence of the optimization problem (2.6).
Notice that, by virtue of Korn’s inequality combined with a Poincar´e inequality (see [4]), the standard [H1(Ωa)]3-norm is equivalent to the norm · εinduced by the inner product
f,gε=
Ωε(f) :ε(g)dx,
wheref andgdenote two regular vector fields having the same dimension as Ω.
The sequence (j(un))n∈N is bounded, and thus there existsu∈[H1(Ωa)]3 such that
unL
2(Ω)
→ u andunH
1(Ω)
u as n→+∞.
Moreover,
u ε≤lim inf
n→+∞ un ε,
which proves the existence of a minimizer. To characterize it, we will use the standard De Rham’s lemma (see, e.g., [36]), stipulating that to take into account the divergence pointwise constraints, it is enough to work with test functions chosen in the space
V(Ω) ={w∈ D(Ωa);∇ ·w= 0 in Ωa},
where D(Ωa) stands for the set of C∞(Ω)-functions having compact support in Ωa. Letw∈ V(Ωa). One has, by virtue of a well-known integration by parts formula (see formula (A.9)),
limt→0
j(u+tw)−j(u)
t =
Ω
ε(u) :ε(w)dx
=−μ
Ω
(Δu+∇(∇ ·u))·wdx.
The use of De Rham’s lemma proves the existence ofp∈L2(Ωa) such thatusatisfies equation−μΔu+∇p= 0, in the sense of distributions. The boundary conditions are derived as usual, using the variational formulation obtained thanks to De Rham’s lemma.
2.2. An abstract shape existence result in dimension 3. To prove an existence result, we need to make the class of admissible domains precise. We denote byD the cylindrical box defined byD={(x1, x2, x3)∈R3|x21+x22≤R20, 0≤x3≤ L}, withR0>0. Imposing some kind of regularity condition is a very classical feature in shape optimization, since these problems are often ill-posed; see [1, 16]. We will consider quasi-open sets included in the string B={(x1, x2, x3)∈R3|0≤x3≤L}.
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Let us recall that a subset Ω⊂Dis said to be quasi open if there exists a nonincreasing sequence of open sets (ωn)n∈Nsuch that
n→+∞lim cap(ωn) = 0 and ∀n∈N, Ω∪ωn is an open set,
where cap denotes the standard capacity defined for compact or open sets (see, e.g., [5, 16]). We fixE⊂ {x3= 0}, a disk whose center is crossed by the axis{x1=x2= 0}, and define
O={Ω quasi open included inD;
∃w∈[H1(D)]3 with Ω ={w= 0}, w|E = (0,0, u0) and cap
Ω∩ {x3=L} = 0 . Theorem 2.3. The problem
(2.7)
⎧⎪
⎪⎪
⎨
⎪⎪
⎪⎩
minJ0(Ω) = 2μ
Ω
|ε(uΩ)|2dx, Ω∈ O,
|Ω| ≤V0,
uΩ solution of (2.4)
has (at least) one solution, whose volume may be chosen equal toV0.
Proof. Let (Ωn)n∈N be a minimizing sequence. Let us denote bymthe infimum.
By virtue of Korn’s inequality combined with a Poincar´e-like inequality (see [4]), and since all the admissible domains are contained in a compact set D, we know that un = uΩn is bounded in [H1(D)]3 (indeed, ε(uΩn) is L2-bounded). Consequently, there exists u ∈ [H1(D)]3 such that (un) converges to u up to an extraction, weakly in [H1(D)]3 and strongly in [L2(D)]3. Using a compactness property of the trace in [L2(E)]3, the conditionu= (0,0, u0) is preserved onE. It remains to prove that cap
Ωu∩ {x3=L} = 0. For that purpose, let us use the fact that uΩn is a solution of (2.4). Integrating the “divergence-free” condition on Ωun yields
−
Supp (u0)u0ds=
D∩{x3=L}u3,nds.
Using the convergence results established previously and a compactness property of the trace, one gets that the above equality remains valid for u and hence ensures that Ωu∗ belongs toO.
Then, the quasi-open Ωu ={u = 0} belongs toO. Furthermore, since u = 0 quasi everywhere onD\Ωu, by weakH1-convergence, one has, by Theorem 2.6,
Ωu
|ε(u)|2dx≤lim inf
n→+∞
Ωn
|ε(un)|2dx=m≤
Ωu
|ε(uΩ
u)|2dx≤
Ωu
|ε(u)|2dx, whence the equality of these quantities.
Finally, thanks to the almost everywhere (a.e.) convergence ofuntou, one has also
|Ωu| ≤lim inf
n→+∞|Ωn| ≤V0.
Now, if|Ωu|=V0, thenΩ := Ω u is a solution of (2.7). If|Ωu|< V0, let us consider any quasi-openΩ such that
Ωu ⊂Ω, |Ω|=V0.
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Thanks to Lemma 2.2, rewriting the objective functionJ0under energetic form J0(Ω) = 2μmin{j(u),u∈Hdiv(Ωa)}
shows immediately that J0 is a decreasing function with respect to the inclusion of sets. By monotonicity of this functional, one has
Ω|ε(uΩ)|2≤
Ωu|ε(uΩ
u)|2, which proves thatΩ is a solution of (2.7).
Remark 1. The solution is a priori not unique, and we have to set additional
“physical” constraints on the domain to get an acceptable solution from the physical point of view. That is why we embedded the problem in the class of cylindrical domains so that we may expect (in particular) uniqueness and more regularity of the optimal graph. Moreover, the choice of admissible shapes we will make in the next section will simplify the study of the related optimization problem and characterize quite precisely the solution.
2.3. Symmetry of solutions for Stokes problems in cylindrical domains.
Before regarding the question of the existence for the shape optimization problem with Stokes constraint, over the set of axisymmetric domains, we need to point out some symmetry properties of the solutions of the Stokes system (2.4). We associate to the classical Cartesian coordinates (x1, x2, x3), the cylindrical coordinates, denoted (r, θ, z), defined by
r=
x21+x22, θ=
⎧⎪
⎨
⎪⎩
2 arctan
x2
x1+ x21+x22
if (x1, x2)∈/R−× {0}
π else
andz=x3.
Proposition 2.4. Let a ∈ A∞. Let us assume that u0 depends only on the variable r. Then, the solution(u, p)of (2.4) posed onΩa satisfies the following:
1. u3 andpare functions of the variables r andz. 2. There existsα∈H1(Da)such that
(2.8) u1=α(z, r) cosθ andu2=α(z, r) sinθ.
Proof. See section A.1.
Let us write u3 = u3(r, z), p = p(r, z) and introduce w = (u3, α) = (w1, w2).
As a direct consequence of the above proposition, system (2.4) can be rewritten in cylindrical coordinates as
−μ
Δw+1 r
∂w
∂r − 1 r2
0 w2
+∇p= 0 inDa, (2.9a)
∇ ·w+w2
r = 0 inDa, (2.9b)
w(0, r) = (u0(r),0) a.e. r∈(0, a(0)), (2.9c)
∂w2
∂z +∂w1
∂r
(L, r) = 0,
−p+ 2μ∂w1
∂z
(L, r) =h(r) a.e. r∈(0, a(L)), (2.9d)
∂w1
∂r (z,0) = 0, w2(z,0) = 0, w(z, a(z)) = 0 a.e. z∈(0, L). (2.9e)
The details of this computation are given in the proof of Proposition 2.4 in sec- tion A.1. Furthermore, the existence of a solution for such a system is guaranteed by the following proposition.
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Proposition 2.5. The two dimensional problem (2.9) has a unique solution (wa, pa)in [H1(Da)]2×L2(Da)if a∈ A∞.
Proof. By Theorem 2.1, problem (2.4) has a unique solution (u, p) in [H1(Ωa)]3× L2(Ωa). Using the result stated in Proposition 2.4, we know that the solution (u, p) of (2.4) is writtenu= (w2cosθ, w2sinθ, w1), andp=pwhere (w,p) is a solution of (2.9), which ensures the existence of a solution to (2.9). The uniqueness for problem (2.9) follows immediately from the uniqueness of solutions for problem (2.4).
Notice that this system has to be understood in a variational sense, which will be made precise in Theorem 2.6.
Using these results, it will be useful to rewrite objective function J0, replacing u by (w2cosθ, w2sinθ, w1), where w is a solution of (2.9). If Ω = Ωa, and first assuming thata∈ A∞, we denote byJ(a,w) the new expression ofJ0(Ωa). A simple but tedious computation (similar to the one of Lemma A.1) yields
(2.10)
J(a,w) = 4πμ L
0
a(z)
0
∂w2
∂r 2
+w22
r2 + ∂w1
∂z 2
+1 2
∂w2
∂z +∂w1
∂r 2
rdrdz.
In view of the use of shape optimization techniques, we need to avoid as much as possible taking into account the regularity constraint on the free boundary. That is why we will state in the next section an existence result using the variational formulation of (2.9) after extending the solution to a fixed compact set.
Remark 2. It may be noticed that solving such a Stokes system is equivalent to inverting a fourth order elliptic operator, close to the bilaplacian.
Indeed, since Da is a two dimensional domain, one can introduce the so-called stream functionψ (see, for instance, [36]). Since the divergence-free condition may be rewritten as
∂
∂z(rw1) + ∂
∂r(rw2) = 0 inDa, we are led to defineψ by the relations
⎧⎨
⎩
rw1 =−∂ψ∂r inDa, rw2 =∂ψ
∂z inDa, ψ= 0 on Γa.
A tedious computation shows that (2.9a)–(2.9b) can be rewritten in terms of the function ψ,
∂
∂r 1
r
∂2ψ
∂r2
+ ∂3ψ
∂z2∂r− 1 r3
∂ψ
∂r +1 μ
∂
∂z(rp) = 0 inDa, (2.11)
∂3ψ
∂z3 + ∂
∂r 1
r
∂2ψ
∂r∂z
− 1 μ
∂p
∂r = 0 inDa. (2.12)
In order to make the pressure term vanish, let us form the equation ∂
∂r
1
r(2.11) +
∂
∂z(2.12). One gets (2.13) ∂4ψ
∂z4+1 r
∂4ψ
∂r4+2 r
∂4ψ
∂z2∂r2− 2
r2 + 1 r3
∂3ψ
∂r3 − 2 r2
∂ψ3
∂r2∂r+2 r3
∂2ψ
∂r2 +4 r5
∂ψ
∂r = 0. Moreover, sincew= 0 on Γa,ψsatisfies
ψ= 0 and ∂ψ
∂n = 0 on Γa.
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2.4. Main result: Shape existence in the class of cylindrical domains.
In subsection 2.2, a general shape existence result for three dimensional domains has been stated. Unfortunately, this result is hardly workable, since the boundary of the quasi-open set Ω may be very irregular (roughly speaking, when the boundary of the domain is locally represented by the graph of a function whose regularity is less than W1,p,p≥2). In such a case, it is sometimes possible to define a solution of the Stokes or Navier–Stokes system, as is emphasized in [8, 9, 26, 27, 35].
This section is devoted to the shape existence results for domains enjoying a cylindrical symmetry property. We no longer assume that the free boundary isW1,∞. Leth0:R→RandhL:R→Rbe two given continuous functions.
Let us introduce the class of admissible domainsUp forp≥2 by Up={Da={(z, a(z))|0≤z≤L} |a∈ Up}, (2.14)
whereUp={a∈ Ap | a Lp≤M, h0(a(0)) =hL(a(L)) = 0}, (2.15)
andAp=
a∈W1,p(0, L)|a0≤a(z)≤a1 forz∈(0, L) , (2.16)
whereh0 andhL are chosen to be compatible with the pointwise constraint satisfied by each element ofAp.
It may be noticed that the compact embedding W1,p(0, L)→ C0([0, L]) for the standardL∞-topology yields in particular the existence ofaM > a0such that a ∞≤ aM for anya∈Up. In what follows, we will make the assumption that
∃η >0|[0, η]⊂supp(u0).
We will see that it permits us to neglect the pointwise constrainta≥a0 a.e.
For a givenp≥2, we consider the shape optimization problem (2.17)
minJ0(Ω),
Ω∈ Ocylp ={(rcosθ, rsinθ, z), ∃a∈ Up, 0≤r < a(z), θ∈T, 0< z < L}, where the functional J0(Ω) denotes the dissipated energy defined, in the case of a sufficiently regular domain Ω (for instance, with Lipschitz boundary), by (2.5). Nev- ertheless, in the case where the boundary is not regular enough, we can defineJ0(Ω) using Theorem 2.7 by
J0(Ω) = 2μ inf
u∈[H1(Ω)]3,∇·u=0 u=u0onE
u=0 on Σ
Ω
|ε(u)|2dx.
Another way to define J0(Ω) consists of defining uΩ thanks to a weak formulation, well adapted for the class of axisymmetric domains with respect to the (Oz)-axis. In particular, since any domain Ω∈ Opcyl is contained in the fixed compact set
D={(rcosθ, rsinθ, z), r∈[0, aM], θ∈T, z∈[0, L]},
we have the temptation to write a weak formulation of the Stokes system onDwhose lateral boundary is quite regular.
For that purpose, let us define the Sobolev spaces Hd(Da) =
ϕ= (ϕ1, ϕ2)∈[H1(Da)]2| ∇ ·ϕ+ϕ2
r = 0 onDa
, (2.18)
Hd,0(Da) =
ϕ= (ϕ1, ϕ2)∈[H1(Da)]2| ∇ ·ϕ+ϕr2 = 0 onDa
andϕ= 0 on Γ0∪Γa
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.The following result plays a crucial role in the proof of the existence of solutions for the shape optimization problem (2.17).
Theorem 2.6. Let p ≥ 2 and Ω ∈ Opcyl, associated with a two dimensional domainDa in cylindrical coordinates.
1. Let a ∈ Ap. If wa ∈ Hd(Da) is a solution of (2.9), then the function w defined by
w=
wa inDa, 0 inD\Da
satisfies (2.19)
∀z∈Hd(Da)∩ Cc∞(Da\Γa), 2μ
D
ε2(w) :ε2(z) +w2z2
r2
rdrdz= 0 andw= 0on Γa,w= (u0,0)on Γ0.
2. Conversely, let w∈[H1(D)]2. If there exista∈ Ap andwasuch that w=
wa inDa, 0 inD\Da,
and ifwsatisfies(2.19)for anyϕ∈Hd(Da)∩C∞c (Da\Γa), thenwais solution of (2.9).
Proof. See section A.2.
Boundary conditions are summarized in Figure 2.2.
Fig. 2.2.DomainsDandDa with boundary conditions.
The result stated in Theorem 2.6 combined with the results of section 2.3 (in par- ticular Proposition 2.4) drives us to consider a new shape optimization problem, over axisymmetric domains, directly deduced from the initial general shape optimization problem (2.17). Indeed, let us introduce the problem
(2.20)
⎧⎨
⎩
minJ(a,wa),
wais solution of (2.19), a∈ Up,
whereJ(a,w) is defined by (2.10).
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In fact, keeping in mind that it seems better to write all the integrals on the fixed compact setD, it is possible to extendwaby 0, so that the shape optimization problem becomes
(2.21)
⎧⎨
⎩
minJ(wa),
wais solution of (2.19), a∈ Up,
where
(2.22) J(w) = 4πμ
D
∂w2
∂r 2
+w22 r2 +
∂w1
∂z 2
+1 2
∂w2
∂z +∂w1
∂r 2
rdrdz.
The following theorem constitutes the main result of this section.
Theorem 2.7. Let p≥2. Problem (2.21) has (at least) a solution.
Proof. See section A.3.
Note that, in general, we do not have uniqueness of the minimizer in shape opti- mization (see, e.g., [1, 16]).
Furthermore, for the needs of future numerical computations of the optimal shape, one may neglect the pointwise constrainta≥a0and replace it bya≥0 forz∈(0, L).
Indeed, the result stated in Theorem 2.7 is a bit more general since it may be noticed that, because of the “divergence-free” constraint, one has for ¯z∈(0, L) fixed,
supp(u0)
u0(r)rdr= a(¯z)
0
w1(r,z¯)rdr,
where supp(u0) denotes the support of the datau0 that is assumed to be of strictly positive measure. This identity comes directly from the integration of the “divergence- free” condition on the restriction of the domain Ωabetween the hyperplanez= 0 and z= ¯z. Thus, the optimal graphacannot vanish because otherwise, the above identity would not be guaranteed.
3. Qualitative and quantitative properties of the optimum.
3.1. Shape derivative for problem (2.7) in the case where Ω is regular.
We are now in position to define the derivative of the functional J0 with respect to the shape Ω. In this section, let us assume that Ω has a C2 boundary, to ensure the differentiability of J0 at Ω. Let us consider a regular (C2, for instance) vector field V:R3→R3 with compact support inside the strip{0< x3< L}. Fortsmall enough, we define Ωt= (I+tV)Ω, the image of Ω by a perturbation of identity and f(t) =J0(Ωt). We recall that the shape derivative ofJ0at Ω exists (already mentioned in [11, 19, 20, 17, 28, 32]) and isf(0). We will denote it by dJ0(Ω;V). The shape derivative with respect to the domain is given by the formula (see [10, 16, 29, 34]) (3.1) dJ0(Ω;V) = 4μ
Ω
ε( ˙u) :ε(u)dx+
∂Ω
|ε(u)|2(V·n)ds, where ( ˙u,p˙) is the unique solution in (H1(Ω))3×L2(Ω) of the linear system
(3.2)
⎧⎪
⎪⎪
⎪⎨
⎪⎪
⎪⎪
⎩
−μΔ ˙u+∇p˙= 0 in Ω,
∇ ·u˙ = 0 in Ω,
˙
u=−∂u∂n(V·n) on Σ,
˙
u= 0 onE, σ( ˙u,p˙)·n= 0 onS.
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Remark 3. The pair ( ˙u,p˙) can be seen as the shape derivative of the pair (u, p).
Let us roughly recall some standard definitions and properties on this notion. We refer to the aforementioned literature for further details. Denote by (ut, pt)∈(H1(Ωt))3× L2(Ωt) the unique solution of
(3.3)
⎧⎪
⎪⎪
⎪⎨
⎪⎪
⎪⎪
⎩
−μΔut+∇pt= 0 in Ωt,
∇ ·ut= 0 in Ωt, ut= 0 on Σt, ut=u0 onE, σ(ut, pt)·n=h onS,
with∂Ωt=E∪Σt∪S. Then, since Ω is assumed to beC2, there exists an extension (ut,pt) of (ut, pt) toRN such that the mapping defined fortsmall enough by
Ψ :t→(ut,pt)∈(H1(R3)3×L2(R3)
isC1. Then, ( ˙u,p˙) is the differential oft→Ψ(t) att= 0. One has also dJ0(Ω;V) = lim
t0
J0(Ωt)−J0(Ω)
t .
In general, it is more convenient to work with another expression of the shape derivative and to write it as a distribution with support Σ =∂Ω\(E∪S). For that purpose, we need in general to introduce an adjoint state. Nevertheless, the Stokes operator is self-adjoint, and we will show that the adjoint problem of (2.4) is the same.
Proposition 3.1. LetΩ∈ O, a domain withC2boundary. Then, the functional J0 is shape-differentiable at Ω, and one has
(3.4) dJ0(Ω;V) =−2μ
∂Ω
|ε(u)|2(V·n)ds.
As a consequence, the first order optimality conditions for problem (2.7) are written as
(3.5) Ωsolution of problem (2.7)withC2 boundary ⇒ |ε(u)|2=constant onΣ. Note that this result is close to those stated in [31, Corollary 1] and [18, sec- tion 3.3].
Proof. The differentiability has been already studied, as mentioned above. Let us now prove (3.4). Since the boundary Σ is assumed to be C2,ε(u) belongs toL2(Σ), and the normal vector n exists everywhere. Let us multiply (2.4) by ˙u and then integrate by parts. We get
2μ
Ω
ε( ˙u) :ε(u)dx=
∂Ω
σ(u, p)n·ud˙ s
=
Σ
pn·∂u
∂n(V·n)ds−2μ
Σ
ε(u)n· ∂u
∂n(V·n)ds.
Sinceuis “divergence-free” and vanishes on Σ, we have on this boundary
• ∂u∂n·n= 0;
• ε(u)n·∂u∂n =|ε(u)|2.
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These remarks permit us to immediately recover the expression of the shape derivative of J0 given in (3.4), using in particular (3.1). Now let us write the first order optimality conditions. Because of the volume constraint, there exists a Lagrange multiplier λ∈R+ such that
−2μ
∂Ω
|ε(u)|2(V·n)ds=−2μλ
Σ
(V·n)ds.
Since this equality holds for a generic elementV∈C2(R3,R3) with compact support inside the strip{0< x3< L}, the first order optimality conditions (3.5) follow.
In the case where the two disks E and S have the same radius R > 0, the prescribed volume is equal to that of the finite cylinder between the two disksEand S;his chosen so that the standard Poiseuille flow1solves (2.4), and if we assume that the flow at the inletEis parabolic, one could naturally think that the cylinder solves the shape optimization problem (2.7). Indeed, in this case, one has forc <0,
u0=c(x21+x22−R2), ε(u) =
⎛
⎝ 0 0 cx1
0 0 cx2
cx1 cx2 0
⎞
⎠, |ε(u)|2= 2c2R2 on Γ.
The first order optimality conditions (3.5) are then satisfied. Nevertheless, the cylin- der is not optimal for this particular choice ofh, as emphasized in [18, Theorem 2.5].
Furthermore, let us notice that the optimal domain Ω is a solution of an overdeter- mined system, that is,
⎧⎪
⎪⎪
⎪⎪
⎪⎨
⎪⎪
⎪⎪
⎪⎪
⎩
−μΔu+∇p= 0 in Ω,
∇ ·u= 0 in Ω, u= 0 on Σ,
|ε(u)|2= constant on Σ, u=u0 onE, σ(u, p)·n=h onS.
The question of the determination of Ω (or some qualitative properties of Ω) is closely linked to the question of the unique continuation property for the Stokes system.
Indeed, the optimal domain Ω satisfies a Cauchy system near the “free boundary” Σ.
It may be noticed that, in dimension 2, a positive answer was given in [30], but for generic domains only.
3.2. Shape derivative for the axisymmetric problem (2.21) in the cases p ∈ [2,+∞] and a ∈ C2([0, L]). Let us define the derivative of the functionalJ with respect to the shape represented by a. It is well known [1, 16] that assuming a ∈ Ap ensures the differentiability of J at wa, and the fact that a is assumed furthermore to be inC2([0, L]) justifies thatwbelongs to [W2,2(Da)]2and permits us to use an integration by parts formula. Let us considerδa∈C2([0, L]), a perturbation ofawith compact support, support which meets neither{z= 0} nor{z=L}. Fort small enough, we defineat=a+tδa, the image ofaby a perturbation of the identity andf(t) =J(wat). We recall that the shape derivative ofJ atwaexists and isf(0).
In what follows, we will prefer the notation J(a) to designate the functionalJ(wa);
similarlyf(0) will be denoted (with a slight abuse of notation) by dJ(a;δa). Let us
1The Poiseuille flow is such thatu= (0,0, u0(r)) for (z, r)∈(0, L)×(0, R0) andpis an affine function of the variablez.
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first define the derivative of the state equation, using the classical calculus of variation results. The shape derivative ofwa denoted ˙wis the solution inHd(Da) of the linear system, written under variational form
(3.6) ∀z∈Hd(Da)∩ Cc∞(Da\Γa), 2μ
Da
ε2( ˙w) :ε2(z) +w˙2z2
r2
rdrdz= 0, and ˙w=−√δa(z)
1+a2(z)
∂wa
∂n on Γa, w˙ = (0,0) on Γ0. Indeed, coming back to the three dimensional representation of our shape optimization problem, following the notations summed up on the left-hand side of Figure 2.1, it is easy to see that on Σa,
n= 1
1 +a2(z)
⎛
⎝ cosθ sinθ
−a(z)
⎞
⎠,
and the perturbation field applied at any point of the boundary ΣaisV= ( δaδacossinθθ 0
).
It is very common in shape optimization to differentiate a Dirichlet boundary condi- tion with respect to the domain (see, e.g., [5, Chapter 4], [16, Chapter 5], or [34]), and one has
˙
w=−∂wa
∂n (V·n) on Γa,
providing the expected result. Note that shape differentiability for problems with Navier–Stokes fluids has been investigated using deformation techniques and the min- max derivation principle in [13]. Using relation (A.10), we claim that the shape derivative with respect to the domain is given by the formula
dJ(a;δa) = 8πμ
Da
!
ε2( ˙w) :ε2(wa) +w˙2wa,2
r2
"
rdrdz+ 4πμ
Γa
|ε2(wa)|2(V2·n2)dσ
= 8πμ
∂Da
ε2(wa)n2·wd˙ σ+ 4πμ
Γa
|ε2(wa)|2(V2·n2)dσ, wheren2=√ 1
1+a2(c−a(z)
1 ),V2= (δa(z)c0 ), anddσstands for the curvilinear measure on Γa, that is, dσ=
1 +a2(z)dz.
Now, since ˙w=−∂w∂na(V2·n2) andε2(wa)n2·∂w∂na =|ε2(wa)|2 (due to the fact thatwavanishes on Γa), one gets
dJ(a;δa) =−4πμ
Γa
|ε2(wa)|2(V2·n2)dσ.
To rewrite this shape derivative as an integral with respect to the variable z, we use the previous expressions ofV2 andn2. This is done in the following proposition.
Proposition 3.2. Leta∈C2([0, L]). Then, the functionalJ is shape-differenti- able atwa and one has for everyδa∈C2([0, L])
(3.7) dJ(a;δa) =−4πμ L
0
|ε2(wa)|2δa(z)dz.
As a consequence, if a∗ ∈C2([0, L]) is the optimal solution to problem (2.21), then for any a∈ Up∩C2([0, L]),
(3.8)
L
0
|ε2(wa∗)|2(a(z)−a∗(z)) dz≤0.
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3.3. Explicit solution of problem (2.21). This section is devoted to the proof of a useful property of the functionalJ that will appear essential to exploring the necessary first order optimality conditions for problem (2.21). In particular, we will give the explicit solution of this optimization problem in the caseh0=hL = 0, where no condition is imposed on the inlet and outlet.
For that purpose, let us use the natural order relation≤defined for two elements of the setAp,p∈[2,+∞] by
a≤b⇐⇒a(z)≤b(z) forz∈(0, L).
Proposition 3.3. The functionala∈ Ap→J(a)is a strictly decreasing function with respect to the order relation ≤.
Proof. The fact thatJ is monotone decreasing is easy to see. It suffices to adapt the end of the proof of Theorem 2.3, noticing that
J(a) = 2μ min
w∈H2,div(Da)
Da
!
|ε2(w)|2+w22
r2
"
rdrdz, whereH2,div(Da) ={w∈[W1,2(Da)]2|w satisfies (A.7), w|
Γ0 = (u0,0), andw|Γ
a = 0}. The monotonicity of J then follows from the inclusion of the Sobolev spaces:
a≤b=⇒H2,div(Da)⊂H2,div(Db).
It remains to prove the strict character of the decreasing property for J. Let us argue by contradiction, considering a ∈ Ap and assuming that there exists b ∈ Ap
such thata≤b andJ(b) =J(a). Let us denote bywathe unique (see Lemma 2.2) minimizer that realizes J(a) and by wb the unique minimizer that realizes J(b).
We will again designate by wa the extension by 0 of this function to the whole domain Db. In particular, it is easy to see that this extension by continuity belongs to [W1,2(Db)]2 and again satisfies (A.7). Hence, because of the uniqueness of the minimizer (underlined in Lemma 2.2), necessarily wa =wb. It implies that at the same time, the minimizerwbis a solution of a Stokes system and vanishes in an open ball included inDb\Da. By virtue of the analyticity of the Stokes operator (see, for instance, [22]), this is absurd, and one deduces that the inequality betweenJ(a) and J(b) is strict.
A direct consequence of Proposition 3.3 is the fact that our shape optimization problem with Stokes constraints rewrites as a purely geometrical problem. As a result, we immediately deduce from this result the following corollary.
Corollary 3.4. In the caseh0=hL= 0, the optimal solution of problem (2.21) is
a∗=a1.
Proof. Proposition 3.3 yields that the unique solution to maxJ(a),
a∈ Ap
isa∗=a1. Sincea∗ is feasible for problem (2.21), the conclusion follows.
Let us now investigate the case where p∈ [2,+∞] and the inlet and outlet are prescribed as follows:
(3.9) h0(x) =x−a0andhL(x) =x−aL,
