Theoretical and Numerical aspects for incompressible fluids Part II: Shape optimization for fluids

PascalFreyandYannickPrivat

(Slides by CharlesDapogny, PascalFreyand YannickPrivat)

CNRS & Univ. Paris 6

feb. 2015

Outlines of the lectures

Session 1 :tools for shape optimization in Fluid Mechanics and existence of optimal shapes Session 2 :Existence and optimality condition (shape derivative)

Session 3 :Application in Fluid Mechanics and algorithms

Joining me :yannick.privat@upmc.fr

download the slides :http ://www.ann.jussieu.fr/frey/nm491.html

Outlines of the lectures

Session 1 :tools for shape optimization in Fluid Mechanics and existence of optimal shapes Session 2 :Existence and optimality condition (shape derivative)

Session 3 :Application in Fluid Mechanics and algorithms

Grading :1 written test (2H) + 1 oral examination

Outlines of the lectures

Session 1 :tools for shape optimization in Fluid Mechanics and existence of optimal shapes Session 2 :Existence and optimality condition (shape derivative)

Session 3 :Application in Fluid Mechanics and algorithms

Grading :1 written test (2H) + 1 oral examination

Joining me :yannick.privat@upmc.fr

download the slides :http ://www.ann.jussieu.fr/frey/nm491.html

1 Several examples of shape optimization problems

2 Tools for shape optimization Generalities

The perimeter functional in shape optimization Existence results in shape optimization Derivation with respect to the domain

Shape derivatives using Eulerian and material derivatives : the rigorous ‘difficult’ way A formal, easier way to compute shape derivatives : C´ea’s method

3 Numerics and applications to Fluid Mechanics Reminders in Fluid Mechanics

Shape optimization in Fluid Mechanics Numerical aspects

Isoperimetric problems

The problem of Dido

Dido was the legendary founder of Carthage (Tunisia). When she arrived in 814 B.C. on the coast of Tunisia, she asked for a piece of land. Her request was satisfied provided that the land could be encompassed by an ox-hide.

With a remarkable mathematical intuition, she cut the ox-hide into a long thin strip and used it to encircle the land. This land became Carthage and Dido became the Queen.

Question : What is the closed curve which has the maximum area for a given perimeter ?

Mathematically, this problem is very close to the standard isoperimetric one : Find Ω⊂R²solution of the problem

maximize area(Ω) such that Per(Ω) = 4km.

Back to the problem of Dido. A naive formulation writes : find the plane curve enclosing with the segment joining its extremity the subdomain having a maximal area. On other words, one has to solve forb>a≥0,

sup

y∈E

Z b a

y(x)dx

where

E={y∈Y | Zb

a

p1 +y⁰²(x)dx=`ety(a) =y(b) = 0}

withY, a given functional space (chosen e.g. so that the problem has a solution).

Isoperimetric problems

The proof

Z´enodore(2^ndcentury B.C.) proves the isoperimetric inequality in the particular case where Ω = polygon.

Until the 20^th century : this result is conjectured but not proved.

Steiner(Swiss mathematician of the 19^th century) publishes a proof, but . . .this proof is erroneous !

Weierstrass(German mathematician of the 19^th century) concludes the proof, by using modern tools ofcalculus of variations.

Generalization inR³ orR^N only known since 1960 (geometric measure theory)

Aerodynamic shape optimization of transonic wings

The model

It requires the use of Navier-Stokes modeling due to the strong nonlinear coupling between airfoil shape, wave drag, and viscous effects.

Criterion, unknown

Goal :reduce the drag (reaction of the flow on the wing ; its

component in the direction of flight is the drag proper and the rest is the lift)

A few percent of drag optimization means a great saving on

commercial airplanes ;

Unknown :the shape of the wing

Aerodynamic optimization of airplane

Aerodynamic shape optimization of transonic wings

! For viscous drag theNavier-Stokes equationsmust be used : for a given shapeSof the wing, it yields the velocityuand pressurepof the fluid at every point.

! For a wing with boundarySmoving at constant speedu∞, the force acting on the wing isF~=

Z

S

µ(∇u+ [∇u]^>)−2 3∇ ·u

nds

| {z }

viscous drag/lift = viscous force

−R

Spnds,wherenis the normal toSpointing outside the domain occupied by the fluid, andµthe viscosity of the fluid.

! Shape optimization problem.The drag is the component ofF~parallel to the velocity at infinity. The optimal design problem writes

S∈Oinf_adJ(S) where J(Ω) =F~·u∞ . Choice ofOad (admissible set of shapes) ?

Sis an open connected bounded subset ofR^d (d= 2 ord= 3)

A geometrical constraint such as the volume being greater than a given value in order to avoid that the wing collapses (|S| ≥S0)

An aerodynamic constraint : the lift must be greater than a given value. Else, the wing will not fly.

Problem and modeling

Consider a fluid of viscosityµflowing inside a cannula-shaped pipe/duct. For instance, we look for the optimal shape of a pipeline.

The optimal design problem writes inf

Ω∈O_adJ(Ω) where J(Ω) = 2µ Z

Ω

|ε(u)|²dx , with

µ, the viscosity of the fluid,u the velocity of the fluid at every point (e.g. given by the Navier-Stokes equations),

Oad is the set of admissible shapes, for instance,E (inlet) andS (outlet) are fixed and we look for the lateral boundary such that Ω open connected subset ofR^d with

Contents

1 Several examples of shape optimization problems

2 Tools for shape optimization Generalities

The perimeter functional in shape optimization Existence results in shape optimization Derivation with respect to the domain

Shape derivatives using Eulerian and material derivatives : the rigorous ‘difficult’ way A formal, easier way to compute shape derivatives : C´ea’s method

3 Numerics and applications to Fluid Mechanics Reminders in Fluid Mechanics

Shape optimization in Fluid Mechanics Numerical aspects

A shape optimization problem writes as the minimization of acost(orobjective) function Jof the domain Ω :

Ω∈Oinf_adJ(Ω),

whereOad is a set ofadmissible shapes(e.g. that satisfyconstraints).

In most mechanical or physical applications, the relevant objective functionsJ(Ω) depend on Ω via astateuΩ, which arises as the solution to a PDE posed on Ω (e.g. the linear elasticity system, or Stokes equations).

Various settings for shape optimization (I)

I. Parametric optimization

The considered shapes are described by means of a set of physicalparameters {pi}_i=1,...,N, typically thicknesses, curvature radii, etc...

•

•

• •

•

• • •

• pi

S

• x h(x)

Description of a wing by NURBS ; the parameters of the representation are the control points pi.

A plate with fixed cross-sectionSis parame- trized by its thickness function h: Ω→R.

! The parameters describing shapes are the onlyoptimization variables, and the shape optimization problem rewrites :

min

{p_i}∈P_adJ(p1, ...,pN), wherePad is a set ofadmissible parameters.

! Parametric shape optimization is eased by the fact that it is straightforward to account forvariationsof a shape{pi}_i=1,...,N :

{pi}_i=1,...,N→ {pi+δpi}_i=1,...,N.

! However, the variety of possible designs is severely restricted, and the use of such a method implies an a priori knowledge of the sought optimal design.

Various settings for shape optimization (III)

II. Geometric shape optimization

! The topology (i.e. the number of holes in 2d) of the considered shapes is fixed.

! Theboundary∂Ω of the shapes Ω itself is the optimization variable.

! Geometric optimization allows more freedom than parametric optimization, since no a priori knowledge of the relevant regions of shapes to act on is required.

@⌦

⌦

Optimization of a shape by performing

‘free’ perturbations of its boundary.

III. Topology optimization

In some applications, the suitabletopologyof shapes is unknown, and also subject to optimization.

In this context, it is often preferred not to describe the boundaries of shapes, but to resort to different representations which allow for a more natural account oftopological changes.

For instance : Describing shapes Ω as characteristic functionsχΩ:D→ {0,1}.

⌦

Optimizing a shape by acting on its topology.

Various settings for shape optimization (V)

! A shape optimization process is a combination of :

Aphysical model, most often based on PDE (e.g. the linear elasticity equations, Stokes system, etc...) for describing the mechanical behavior of shapes, Adescriptionof shapes and their variations (e.g. as sets of parameters, density functions, etc...),

Anumerical descriptionof shapes (e.g. by a mesh, a spline representation, etc...)

! These choices are strongly inter-dependent and influenced by the sought application.

! However very different in essence, all these different methods for shape optimization share a lot of common features.

! We are going to focus ongeometric shape optimization methods.

Disclaimer

I This course isveryintroductory, and by no means exhaustive, as well for theoretical as for numerical purposes.

I See the (non exhaustive) References section to go further.

Bibliography/References

Allaire G.,Conception optimale de structures, Math´ematiques et Applications 58, Springer, (2006)

Bendsoe M.P. and Sigmund O.,Topology Optimization, Theory, Methods and Applications, 2nd Edition, Springer, (2003)

Henrot A., and Pierre M.,Variation et optimisation de formes, une analyse g´eom´etrique, Springer, (2005)

Mohammadi B. and Pironneau O.,Applied shape optimization for fluids, Oxford University Press,28, (2001)

Pironneau O., Optimal Shape Design for Elliptic Systems, Springer, (1984)

Sethian J.A.,Level Set Methods and Fast Marching Methods : Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science, Cambridge University Press, (1999).

and many others. . .

1 Several examples of shape optimization problems

2 Tools for shape optimization Generalities

The perimeter functional in shape optimization Existence results in shape optimization Derivation with respect to the domain

Shape derivatives using Eulerian and material derivatives : the rigorous ‘difficult’ way A formal, easier way to compute shape derivatives : C´ea’s method

3 Numerics and applications to Fluid Mechanics Reminders in Fluid Mechanics

Shape optimization in Fluid Mechanics Numerical aspects

General definition of perimeter

Definition : Perimeter in the sense ofDe Giorgi

Let Ω be a (Lebesgue) mesurable set inR^d. Theperimeter ofΩ is defined by Per(Ω) = sup{

Z

Ω

div(ϕ)dx|ϕ∈ D(R^d,R^d), kϕk∞≤1},

whereD(R^d,R^d) is the set ofC^∞functionsϕ:R^d →R^d having a compact support.

Definition introduced by EnnioDe Giorgi(1928-1996) Note that, for a functionϕ∈ D(R^d,R^d), there holds

Z

Ω

div(ϕ)dx= Z

R^d

div(ϕ)χΩ(x)dx=−h∇χΩ, ϕiD⁰,D,

whereχΩ denotes thecharacteristic function ofΩ, in other words χΩ(x) =

1 ifx∈Ω 0 else.

Reminder on the notion of “regular set”

1 An open set Ω ofR^d is said to have aLipschitz boundaryif, for allx0∈∂Ω, there exist a cylinderK=K⁰×]−a,a[ in a local cartesian basis with originx0= 0, with K⁰an open ball inR^d−1with radiusr, and a Lipschitz functionϕ:K⁰→]−a,a[

such thatϕ(0) = 0 and

Ω∩K={(x⁰,y)∈K |y > ϕ(x⁰)} et ∂Ω∩K={(x⁰, ϕ(x⁰))|x⁰∈K⁰}.

2 An open set is said of classC¹by replacing the word ”Lipschitz” byC¹in the definition above.

Proposition (Link with the usual definition of perimeter for regular sets) Let Ω, be a bounded open set of classC¹. Then, Per(Ω) =R

∂Ωdσ, wheredσdenotes the surface element on∂Ω.

General definition of perimeter

Proposition

Let Ω, be a bounded open set of classC¹. Then, Per(Ω) =R

∂Ωdσ, wheredσdenotes the surface element on∂Ω.

Sketch of the proof : let ϕ ∈ D(R^d,R^d) such that kϕk∞ ≤ 1. According to Green’s formula, one has

Z

Ω

div(ϕ)dx=

d

X

i=1

Z

Ω

∂ϕi

∂xi

dx= Z

∂Ω

ϕinidσ= Z

∂Ω

ϕ·νdσ,

whereνis the outward unit normal vector. Using the triangle inequality, one has for every smooth functionϕ,

Z

Ω

div(ϕ)dx≤ Z

∂Ω

|ϕ||ν|dσ≤ kϕk∞

Z

∂Ω

dσ≤ Z

∂Ω

dσ.

Passing to the sup, one gets Per(Ω)≤R

∂Ωdσ.

Proposition

Let Ω, be a bounded open set of classC¹. Then, Per(Ω) =R

∂Ωdσ, wheredσdenotes the surface element on∂Ω.

Proof (converse sense) : One would like to chooseϕ=νon∂Ω.

Idea : find a sequence of functions (ϕn)n∈N∈ D(R^d,R^d)^Nto approximateν. IntroduceN, a continuous extension of the normal vectorν. One regularizesN by definingϕn=N ∗ρn, where (ρn)n∈N is a mollifier.

One thus construct a sequence of functions inD(R^d,R^d) converging uniformly toN and

s.t. Z

Ω

divϕndx= Z

∂Ω

ϕn·νdσ−−−−→

n→+∞

Z

∂Ω

ν·νdσ= Z

∂Ω

dσ.

General definition of perimeter

Reminder on Radon measures :recall that iff ∈L¹(R^d,R^d), then by duality kfk_L1=

Z

R^d

|f(x)|dx= sup{

Z

R^d

f(x)·ϕ(x)dx, ϕ∈ D(R^d,R^d)andkϕk∞≤1}.

More generally, this formula makes sense for everyRadon measureµhaving a finite mass (i.e. every linear continuous form on the spaceC0⁰(R^d) of continuous functions having a compact support inR^d)

Moreover, a Radon measureµhas a finite mass onR^d iff the quantity kµk1= sup{hµ, ϕiD⁰,D, ϕ∈ D(R^d,R^d)etkϕk∞≤1}

is finite.

Recall that

Per(Ω) = sup{−h∇χΩ, ϕiD⁰,Dϕ∈ D(R^d,R^d), kϕk∞≤1}

Proposition

Let Ω be a measurable subset ofR^d. Then, the quantity Per(Ω) is finite iff∇χΩis a Radon measure having a finite mass and in that case, Per(Ω) =k∇χΩk1.

Theorem (compactness properties of the perimeter)

Let (Ωn)n∈Nbe a sequence of measurable sets inR^d. Let us assume the existence ofC>0 s.t.

∀n∈N, |Ωn|+ Per(Ωn)≤C.

Then, there exist a measurable set Ω⊂R^d and a subsequence (Ωn_k)k∈Ns.t.

χΩ_nk →χΩdansL¹_loc(R^d) et h∇χΩ_nk, ϕi → h∇χΩ, ϕi, ∀ϕ∈ C₀⁰(R^d).

Moreover, if all the elements of the sequence (Ωn)n∈N belong to an open setD having a finite measure, then the sequence (χΩ_nk)k∈N converges toχΩinL¹(D).

! The convergence of the gradients of the characteristic functions coincides with the weak-∗convergence of the bounded Radon measures.

! The proof of this theorem rests upon the compactness of the injection BV(R^d),→L¹_loc(R^d).

General definition of perimeter

Definition : convergence in the sense of characteristic functions.

Let (Ωn)n∈N and Ω denote respectively a sequence of measurable sets and a measurable set ofR^d. The sequence (Ωn)n∈Nis said toconverge in the sense of characteristic functions ifχΩ_n →χΩ inL¹_loc(R^d).

According to the previous theorem, a sequence of domains contained in a bounded open setDand whose perimeter is uniformly bounded converges (up to a

subsequence) in the sense of characteristic functions.

Let (Ωn)n∈N be a sequence of measurable subsets ofR^d. Without additional assumption, one cannot conclude that (χΩn)n∈N converges up to a subsequence to χΩ. Nevertheless, sinceL^∞(R^d;{0,1}) is the topological dual of the space

L¹(R^d;{0,1}), there exist a subsequence (χΩ_nk)k∈N converging weakly-?inL^∞to a functiona∈L^∞(R^d,[0,1])(Banach-Alaoglu theorem).

Consider d= 1 and the sequence (ωn)n∈N of subsets of ]0, π[ defined by ωn=Sn

k=1 kπ

n+1−_4n^π,_n+1^kπ +_4n^π

,for everyn∈N^∗.

One can prove (exercise) :|ωn|=^π₂, and the sequence (χωn)n∈N converges weakly-∗

in L^∞to the constant functiona(·) = ¹₂.

1 Several examples of shape optimization problems

2 Tools for shape optimization Generalities

The perimeter functional in shape optimization Existence results in shape optimization Derivation with respect to the domain

Shape derivatives using Eulerian and material derivatives : the rigorous ‘difficult’ way A formal, easier way to compute shape derivatives : C´ea’s method

3 Numerics and applications to Fluid Mechanics Reminders in Fluid Mechanics

Shape optimization in Fluid Mechanics Numerical aspects

Existence of solutions in shape optimization

General issues Let us consider the general optimal design problem

Ω∈Oinf_adJ(Ω) with Oad⊂ A(D) ={Ω⊂D, Ωopen},

whereD is an open set ofR^d.

Study plan for existence issues

1 is the quantity infΩ∈O_adJ(Ω) finite ?

In that case, introduce a minimizing sequence (Ωn)n∈N of elements ofOad. Moreover, the sequence (J(Ωn))n∈N is bounded.

2 what topology to choose on the admissible set of domainsOadto guarantee : ,→ that one can find a subsequence of (Ωn)n∈N converging to a certain Ω∈ O_adfor this

topology ?

,→ that the functionalJbe lower semi-continuous for this topology, in other words so that Ωn→Ω ⇒ J(Ω)≤lim inf

n→+∞J(Ωn) Consequence : the problem above has a solution.

In the context of fluid mechanics :we have to deal with PDE constraints ! !

A toy problem for PDE constraints LetD a bounded connected open set andOadbe a subset of

A(D) ={Ω open, Ω⊂D}.

We consider the shape optimization problem

Ω∈Oinf_adJ(Ω) where J(Ω) = Z

Ω

(uΩ(x)−u0(x))²dx,

withu0∈L²(D) given, anduΩthe unique solution of the P.D.E.

−∆uΩ=f x∈Ω uΩ= 0 x∈∂Ω, withf ∈H⁻¹(D).

To obtain the existence of a solution, we will have, in addition to the convergence of the minimizing sequence (Ωn)n∈N in a sense to make precise, to obtain the convergence of (uΩ_n)n∈N inL²(D) for example.

Existence of solutions in shape optimization

Definition (Hausdorff complement topology)

Letx∈R^d andE a closed subset ofR^d. We denote

dE(x) = dist(x,E) := inf{|x−y|, y ∈E}.

Hausdorff distance between two closed sets :forA,B, two closed sets ofR^d, dH(A,B) = max{sup

x∈A

dB(x),sup

x∈B

dA(x)}.

Convergence of open sets in the sense of Hausdorff (Hausdorff complementary topology) :forAandB inA(D),

dH^c(A,B) =dH(D\A,D\B).

As a consequence, we will say that (Ωn)n∈N (sequence of A(D)) converges to Ω in the sense of HausdorffifdH^c(Ωn,Ω)→0 asn→+∞.

Remarks on Hausdorff convergence

Very important property

For everyA,B closed subsets ofD⊂R^d, dH(A,B) =kdA−dBk_C0(D):= sup

x∈D

|dA(x)−dB(x)|.

Proof :there holds

kdA−dBk_C0(D)≥max{kdA−dBk_C0(A),kdA−dBk_C0(B)} ≥dH(A,B).

Let us show the converse inequality. Letx∈B andy ∈A. By compactness, there existxB ∈B s.t.dB(x) =|x−xB|. Hence,

dA(x)−dB(x)≤ |y−x| − |x−xB| ≤ |y−xB| ≤dA(xB)≤sup

x∈B

dA(x).

By inverting the roles played byAandB, we infer the desired inequality.

Existence of solutions in shape optimization

Back to the Hausdorff complement topology

Proposition

1 LetAandB inD. One hasdH(A,B) = 0 iffA=B.

2 Introduce the space of ”distance” functions of the subsets ofD

Cd(D) ={dΩ, withΩ6=∅andΩ⊂D} with dΩ:x7→d(x,Ω).

Then,Cd(D) is a compact subspace ofC⁰(D).

! It is enough to restrict the set of admissible shapes to classes of open sets. Indeed, {dΩ^c, Ω⊂D andΩ6=∅}={dΩ^c, Ω open set included inD}.

To show this equality, consider Ω⊂D s.t. Ω6=∅.

Consider the open setO= Ω^cc. SinceO^c= Ω^c, there holdsdO^c=d_Ωc=dΩ^c. Therefore, the proposition above must be interpreted in the following sense : if (Ωn)n∈Nis a sequence ofA(D) converging to ˜Ω in the sense of Hausdorff, then the limits of the sequence (Ωn)n∈N belong to the same equivalence class and there exists Ω∈ A(D) s.t. (Ωn)n∈N converges to Ω.

Proposition (compactness and l.s.c. for the topologyH^c)

1 The setA(D) endowed with the metricdH^c is compact.

2 Let (Ωn)n∈N, a sequence of elements inA(D) that converges in the sense of Hausdorff to Ω. For every compact subsetK⊂Ω, there exists an integerN(K)∈N such thatK⊂Ωnfor everyn≥N(K).

3 The Lebesgue measure is lower semi-continuous for the topologydH^c. Examples (exercise)

Consider an open setD and (xn)n∈N, a sequence of dense points inD. Set Ωn=D\{xk}1≤k≤n. Then, (Ωn)n∈N converges in the sense of Hausdorff to∅.

Ωn=]0,1[\{_n¹}n∈N converges in the sense of Hausdorff to ]0,1[.

The uniform cone property

Definition

Lety, a point ofR^d,ξ, a unit vector andε >0.

1 Introduce the coneC(y, ξ, ε) with apexy, directionξand dimensionε, defined by C(y, ξ, ε) ={z∈R^d| hz−y, ξi_Rd ≥cosεkz−yk_Rd and 0<kz−yk_Rd < ε}, whereh·,·i_Rd is the euclidean scalar product ofR^d andk · k_Rd, the associated euclidean norm.

2 An open set Ω verifies theε-cone property if for everyx ∈Ω, there existsξx, a unit vector such that :

∀y ∈Ω∩B(x, ε), C(y, ξx, ε)⊂Ω, whereB(x, ε) denotes the open ball with centerx and radiusε.

! Note that, according to the definition ofC(y, ξ, ε),y ∈/C(y, ξ, ε).

! Theorem: an open bounded set Ω satisfies theε-cone property iff it has a Lipschitz boundary.

C(y, ξx, ε) y x

ε

ξx

Illustration of theε-cone property

Left : set verifying the ε- cone property,

Right : set that does not verify theε-cone property

The uniform cone property

Compactness property ofε-cone property

Proposition

Let ε > 0 and (Ωn)n∈N, a sequence of open sets contained in a bounded open set D satisfying theε-cone property for someε >0.

Then, there exist an open set Ω and a subsequence (Ωn_k)k∈Nconverging to Ω in the sense of Hausdorff. Moreover,

Ω has theε-cone property,

Ωn_k and∂Ωn_k converge in the sense of Hausdorff to Ω and∂Ω.

one may assume that (Ωn_k)k∈N also converges to Ω in the sense of characteristic functions.

Example 1 : an isoperimetric like problem

Ω∈Oinf_V

0

Per(Ω) avec OV0={Ω⊂D| Z

Ω

f(x)dx=V0},

withD, a bounded open set ofR^d,f ∈L¹_loc(R^d) andV0>0 given.

Theorem

The problem above has (at least) a solution.

Existence issues : two examples

Theorem

The problem above has (at least) a solution.

Proof : let (Ωn)n∈N be a minimizing sequence in OV₀. Then, there exists c > 0 s.t.

Per(Ωn)≤c for alln∈N.

Since Ωn⊂Dfor alln∈N, one infers by compactness of sets having a bounded perimeter that :∃(Ωn_k)k∈N and a measurable set Ω s.t.χΩ_nk →χΩinL¹_loc(R^d). Therefore,

Z

Ω_nk

f(x)dx= Z

R^d

χΩ_nk(x)f(x)dx−−−−→

n→+∞

Z

Ω

f(x)dx.

Letϕ∈ D(R^d,R^d). Hence, Z

Ω

div(ϕ)dx = lim

n→+∞

Z

Ω_nk

div(ϕ)dx≤lim inf

k→+∞ sup

ϕ∈D(R^d,R^d), kϕk_∞≤1

Z

Ω_nk

div(ϕ)dx

= lim inf

k→+∞Per(Ωn_k).

Passing to the sup in this inequality yields Per(Ω)≤lim infk→+∞Per(Ωn_k). The conclusion follows.

Example 2 : back to the toy problem

Letε >0 andD be a bounded open set ofR^d.

Ω∈Oinf_ad,εJ(Ω) where J(Ω) = Z

Ω

(uΩ(x)−u0(x))²dx, where

Oad,ε={Ω∈ A(D)| |Ω| ≤1 and Ω satisfies theε-cone property}, withu0∈L²(D) given, anduΩthe unique solution of the P.D.E.

−∆uΩ=f x∈Ω uΩ= 0 x∈∂Ω, withf ∈H⁻¹(D).

Theorem

The problem above has (at least) a solution.

Existence issues : two examples

Theorem

The problem above has (at least) a solution.

Proof :let (Ωn)n∈N be a minimizing sequence inOad,ε. Then, there exists a subsequence (Ωn_k)k∈N such that

(Ωn_k)k∈N converges to Ω∈ A(D) for the Hausdorff complement topology and in the sense of characteristic functions ;

Ω satisfies theε-cone property ;

|Ω| ≤lim infk→+∞|Ωn_k| ≤1 (the measure is l.s.c. for the Hausdorff topology) ; Multiply−∆uΩ_nk =f byuΩ_nk and integrate by parts. It yields

kuΩ_nkk_H1

0(D)≤ hf,uΩ_nki_H−1,H₀¹≤ kfk_H−1(D)kuΩ_nkk_H1 0(D).

Thus, (uΩ_nk)kis bounded inH¹(D) and according to Rellich-Kondratov theorem, it converges tou^∗∈H0¹(D) weakly inH¹and strongly inL².

Theorem

The problem above has (at least) a solution.

Proof :

(uΩ_nk)k∈N is bounded inH¹(D) and according to Rellich-Kondratov theorem, it converges tou^∗∈H0¹(D) weakly inH¹(D) and strongly inL²(D).

Passing to the limit in the variational formulation

∀ϕ∈H0¹(Ω), Z

D

∇uΩ_nk · ∇ϕdx=hf, ϕi_H−1,H₀¹

proves thatu^∗satisfies−∆u^∗=f in a distributional sense.

WritinguΩ_nk(x)(χΩ_nk(x)−χD(x)) = 0 for almost everyx ∈D. Uand passing to the limit yieldsu^∗= 0 almost everywhere inD\Ω^∗.

Consequently,u^∗∈H₀¹(Ω^∗) and thenu^∗solves the P.D.E.

−∆u^∗=f x∈Ω u^∗= 0 x∈∂Ω, Conclusion :infJ= limk→+∞J(Ωn ) =R

(u^∗−u0)²dxwhence the existence.

Contents

1 Several examples of shape optimization problems

2 Tools for shape optimization Generalities

The perimeter functional in shape optimization Existence results in shape optimization Derivation with respect to the domain

Shape derivatives using Eulerian and material derivatives : the rigorous ‘difficult’ way A formal, easier way to compute shape derivatives : C´ea’s method

3 Numerics and applications to Fluid Mechanics Reminders in Fluid Mechanics

Shape optimization in Fluid Mechanics Numerical aspects

Hadamard’s boundary variation method describes variations of a reference, Lip- schitz domain Ω of the form :

Ω7→Ωθ:= (I+θ)(Ω), for ‘small’ vector fields θ ∈ W^1,∞ R^d,R^d

.

⌦

•

•

x•

(x)

⌦✓

Lemma

Forθ∈W^1,∞(R^d,R^d)such that||θ||_W1,∞(R^d,R^d)<1, the application(I+θ)is a Lipschitz diffeomorphism.

Differentiation with respect to the domain : Hadamard’s method

Definition

Given a smooth domain Ω, a scalar function Ω7→J(Ω)∈Ris said to beshape differentiableat Ω if the function

W^1,∞

R^d,R^d

3θ7→J(Ωθ)

is Fr´echet-differentiable at 0, i.e. the following expansion holds in the vicinity of 0 : J(Ωθ) =J(Ω) +J⁰(Ω)(θ) +o

||θ||_W1,∞(R^d,R^d)

.

The linear mappingθ7→J⁰(Ω)(θ) is the shape derivativeofJ at Ω.

Idea : The shape derivativeJ⁰(Ω)(θ) of a ‘regular’ functionalJ(Ω) only depends on the normal componentθ·nof the vector fieldθ.

⌦

⌦✓ ✓

Figure:At first order, atangentialvector fieldθ, (i.e.θ·n= 0) only results in aconvectionof the shapeΩ, and it is expected that J⁰(Ω)(θ) = 0.

Structure of shape derivatives (II)

Lemma

LetΩbe a domain of classC¹. Assume that the application C^1,∞(R^d,R^d)3θ7→J(Ωθ)∈R

is of classC¹. Then, for any vector fieldθ∈ C^1,∞(R^d,R^d)such thatθ·n= 0on∂Ω, one has : J⁰(Ω)(θ) = 0.

Corollary

Under the same hypotheses, ifθ1, θ2∈ C^1,∞(R^d,R^d)have the same normal component, i.e.θ1·n=θ2·n on∂Ω, then :

J⁰(Ω)(θ1) =J⁰(Ω)(θ2).

Actually, the shape derivatives of ‘many’ integral objective functionalsJ(Ω) can be put under the form :

J⁰(Ω)(θ) = Z

∂Ω

vΩ(θ·n)ds,

wherevΩ:∂Ω→Ris a scalar field which depends onJand on the current shape Ω.

This structure lends itself to the calculation of adescent direction: lettingθ=−tvΩn, for a small enoughdescent stept>0 yields :

J(Ωtθ) =J(Ω)−t Z

∂Ω

vΩ²ds+o(t)<J(Ω).

First examples of shape derivatives

Theorem

Let Ω ⊂ R^d be a bounded Lipschitz domain, and f ∈ W^1,1(R^d) be a fixed function.

Consider the functional :

J(Ω) = Z

Ω

f(x)dx;

thenJis shape differentiable at Ω and its shape derivative is :

∀θ∈W^1,∞(R^d,R^d), J⁰(Ω)(θ) = Z

∂Ω

f (θ·n)ds.

⌦_✓ ⌦

•

•

•

x+✓(x) x

Figure:Physical intuition : J(Ωθ)is obtained from J(Ω)by adding the blue area, where θ·n>0, and removing the red area, whereθ·n<0. The process is ‘weighted’ by the integrand function f .

First examples of shape derivatives

Remarks :

This result is actually a particular case of theTransport(orReynolds)theorem, used to derive the equations of conservation from conservation principles.

It allows to calculate the shape derivative of the volume functional Vol(Ω) =R

Ω1dx:

∀θ∈W^1,∞(R^d,R^d), Vol⁰(Ω)(θ) = Z

∂Ω

θ·n ds= Z

Ω

div(θ)dx.

In particular, ifdiv(θ) = 0, the volume does not vary (at first order) when Ω is perturbed byθ.

Proof :The formula proceeds from a change of variables : J(Ωθ) =

Z

(I+θ)(Ω)

f(x)dx= Z

Ω

|det(I+∇θ)|f ◦(I+θ)dx.

•The mappingθ7→det(I+∇θ) is Fr´echet differentiable at 0, and : det(I+∇θ) = 1 +div(θ) +o(θ), o(θ)

||θ||_W1,∞(R^d,R^d) −→

θ→00.

•Iff ∈W^1,1(R^d),θ7→f ◦(I+θ) is also Fr´echet differentiable at 0 and : f ◦(I+θ) =f +∇f ·θ+o(θ).

•Combining those three identites and Green’s formula lead to the result.

First examples of shape derivatives

Theorem

Let Ω0⊂R^dbe a bounded,regular enoughdomain, andg ∈W^2,1(R^d) be afixedfunction.

Consider the functional :

J(Ω) = Z

∂Ω

g(x)ds;

thenJis shape differentiable at Ω0and its shape derivative is : J⁰(Ω)(θ) =

Z

∂Ω

∂g

∂n+κg

(θ·n)ds, whereκstands for themean curvatureof∂Ω.

Example : The shape derivative of theperimeterP(Ω) =R

∂Ω1ds is : P⁰(Ω)(θ) =

Z

∂Ω

κ(θ·n)ds.

The examples of physical interest are those ofPDE constrained shape optimization, i.e.

one aims at minimizing functions which depend on Ω via the solutionuΩof a PDE posed on Ω, for instance (in most of the forthcoming examples) :

J(Ω) = Z

Ω

j(uΩ)dx+ Z

∂Ω

k(uΩ)ds,

whereuΩis e.g. the solution to the linear elasticity system posed on Ω, andj,k are given functions.

Doing so borrows methods from optimal control theory (adjoint techniques, etc...)

The framework

Henceforth, we rely on the model of theLaplace equation: the stateuΩis solution to the system







−∆u=f in Ω

u= 0 on∂Ω (Dirichlet B.C)

∂u

∂n = 0 on∂Ω (Neumann B.C) whereR

Ωf dx= 0in theNeumanncase.

The associated variational formulation reads :

∀v ∈H₀¹(Ω)/H¹(Ω), Z

Ω

∇u· ∇v dx− Z

Ω

fv dx= 0.

We aim at calculating the shape derivative ofJ(Ω) =R

Ωj(uΩ)dx, wherej:R→R is a ‘smooth enough’ function.

Contents

1 Several examples of shape optimization problems

2 Tools for shape optimization Generalities

The perimeter functional in shape optimization Existence results in shape optimization Derivation with respect to the domain

Shape derivatives using Eulerian and material derivatives : the rigorous ‘difficult’ way A formal, easier way to compute shape derivatives : C´ea’s method

3 Numerics and applications to Fluid Mechanics Reminders in Fluid Mechanics

Shape optimization in Fluid Mechanics Numerical aspects

Eulerian and Lagrangian derivatives (I)

The rigorous way to address this problem requires a notion of differentiation of functions Ω7→uΩ, which to a domain Ω associate a function defined on Ω. One could think of two ways of doing so :

The Eulerian point of view : For a fixed x ∈ Ω, u⁰Ω(θ)(x) is the derivative of the application

θ7→uΩ_θ(x).

x•

⌦

⌦✓

u⌦✓

u⌦

The Lagrangian point of view : For a fixedx ∈Ω, ˙uΩ(θ)(x) is the derivative of the application

θ7→uΩ_θ((I+θ)(x)).

x•

⌦

⌦✓

✓(x)

•x+✓(x)

The Eulerian notion of shape derivative, however more intuitive, is more difficult to define rigorously. In particular, differentiating theboundary conditionssatsified byuΩ

is clumsy :

even forθ‘small’, uΩθ(x)may not make any sense if x∈∂Ω!

The Lagrangian notion of shape derivative can be rigorously defined, and lends itself to mathematical analysis.

The Eulerian derivative will bedefinedafter the Lagrangian derivative, from the formal use of chain rule over the expressionu(I+θ)(Ω)◦(I+θ) :

∀x ∈Ω, u˙Ω(θ)(x) =uΩ⁰(θ)(x) +∇uΩ(x)·θ(x).

Eulerian and Lagrangian derivatives (III)

Let Ω7→u(Ω)∈H¹(Ω) be a function which to a domain, associates a function on the domain.

Definition

The functionu: Ω7→u(Ω) admits amaterial, orLagrangianderivative ˙u(Ω) at a given domain Ω provided thetransported function

W^1,∞(R^d,R^d)3θ7−→u(θ) :=u(Ωθ)◦(I+θ)∈H¹(Ω),

which is defined in the neighborhood of 0∈W^1,∞(R^d,R^d), is differentiable atθ= 0.

We are now in position todefine the notion of Eulerian derivative.

Definition

The functionu: Ω7→u(Ω) admits aEulerian derivativeu⁰(Ω)(θ) at a given domain Ω in the directionθ∈W^1,∞(R^d,R^d) if it admits a material derivative ˙u(Ω)(θ) at Ω, and

∇u(Ω)·θ∈H¹(Ω).

One defines then :

u⁰(Ω)(θ) = ˙u(Ω)(θ)− ∇u(Ω).θ∈H¹(Ω). (1)

Eulerian and Lagrangian derivatives (V)

Proposition

Let Ω⊂R^d be a smooth bounded domain, and suppose that Ω7→u(Ω) has aLagrangian derivativeu(Ω) at Ω. If˙ j:R→Ris regular enough, the functionJ(Ω) =R

Ωj(u(Ω))dx is thenshape differentiableat Ω, and :

J⁰(Ω)(θ) = Z

Ω

˙

u(Ω)(θ)j⁰(u(Ω)) +j(u(Ω)) div(θ) dx.

for everyθ∈W^1,∞(R^d,R^d).

Ifu(Ω) has aEulerian derivativeu⁰(Ω) at Ω, one has the‘chain rule’: J⁰(Ω)(θ) =

Z

∂Ω

j(u(Ω))θ·n ds

| {z }

Derivative of Ω7→R Ωj(uΩ ) with respect to its first argument

+ Z

Ω

j⁰(u(Ω))u⁰(Ω)(θ)dx

| {z }

Derivative of Ω7→R Ωj(uΩ ) with respect to its second argument

.

Practically speaking, one mainly uses this last expression of theshape derivative.

Let us return to our problem of calculating the shape derivative of : J(Ω) =

Z

Ω

j(uΩ)dx, where

−∆u=f in Ω u= 0 on∂Ω .

The following result characterizes theLagrangian derivativeof Ω7→uΩ. Its proof can be adapted to many different PDE models :

Theorem

Let Ω⊂R^d be a smooth bounded domain. The application Ω7→uΩ ∈H₀¹(Ω) admits a Lagrangian derivativeu˙Ω(θ), and for anyθ∈W^1,∞(R^d,R^d), ˙uΩ(θ)∈H₀¹(Ω) is the unique solution to :

−∆Y =−∆(θ· ∇uΩ) in Ω

Y = 0 on∂Ω .

Eulerian and Lagrangian derivatives (VII)

Idea of the proof :The variational problem satisfied byuΩ_θ is :

∀v ∈H0¹(Ωθ), Z

Ω_θ

∇uΩ_θ· ∇v dx = Z

Ω_θ

fv dx.

By a change of variables, thetransported functionu(θ) :=uΩ_θ◦(I+θ) satisfies :

∀v ∈H₀¹(Ω), Z

Ω

A(θ)∇u(θ)· ∇v dx = Z

Ωθ

|det(I+∇θ)|fv dx, whereA(θ) :=|det(I+∇θ)|∇θ∇θ^T.

This variational problem features afixed domainand afixed function spaceH0¹(Ω), and only the coefficients of the formulation depend onθ.

The problem can now be written as an equation foru(θ) : F(θ,u(θ)) =G(θ), for appropriate definitions of the operators :

F:W^1,∞(R^d,R^d)×H₀¹(Ω)→H⁻¹(Ω), G:W^1,∞(R^d,R^d)→H⁻¹(Ω).

A use of theimplicit function theoremprovides the result.

In particular, theLagrangian derivativeu˙Ω(θ) satisfies :

∀v ∈H0¹(Ω), Z

Ω

∇u˙Ω(θ)· ∇v dx = Z

Ω

∇(θ· ∇uΩ)· ∇v dx.

Eulerian and Lagrangian derivatives (IX)

Remark : TheEulerian derivativeofuΩcan now be computed from itsLagrangian derivative. It satisfies : −∆U= 0 in Ω

U=−(θ·n)^∂u_∂n^Ω on∂Ω , or under variational form :

∀v ∈H0¹(Ω), Z

Ω

∇u⁰Ω(θ)· ∇v dx = Z

∂Ω

∂uΩ

∂n vθ·n ds.

Using this formula in combination with : J⁰(Ω)(θ) =

Z

∂Ω

j(uΩ)θ·n ds+ Z

Ω

j⁰(uΩ)uΩ⁰(θ)dx

will allow to expressJ⁰(Ω)(θ) as a completelyexplicitexpression ofθ: this is theadjoint methodfrom optimal control theory.

Goal of the adjoint method :writeJ⁰(Ω)(θ) under the form J⁰(Ω)(θ) =

Z

∂Ω

something×(θ·n)ds wheresomethingdoes not depend onθ.

Idea :‘lift up’ the term ofJ⁰(Ω)(θ) which features the Eulerian derivative ofuΩ

by introducing anadequate auxiliary problem.

LetpΩ∈H₀¹(Ω) be defined as the solution to the problem : −∆p=−j⁰(uΩ) in Ω

p= 0 on∂Ω . The variational formulation for pΩis :

∀v ∈H0¹(Ω), Z

Ω

∇pΩ· ∇v dx=− Z

Ω

j⁰(uΩ)vdx, ... to be compared with :

J⁰(Ω)(θ) = Z

∂Ω

j(uΩ)θ·n ds+ Z

Ω

j⁰(uΩ)u_Ω⁰(θ)dx.

Eulerian and Lagrangian derivatives (XI) : the adjoint method

Thus,J⁰(Ω)(θ) = Z

∂Ω

j(uΩ)θ·n ds+ Z

Ω

j⁰(uΩ)u_Ω⁰(θ)dx

= Z

∂Ω

j(uΩ)θ·n ds− Z

Ω

∇pΩ·∇u⁰Ω(θ)dx

= Z

∂Ω

j(uΩ)θ·n ds− Z

∂Ω

∂pΩ

∂n

∂uΩ

∂n θ·ndx ,

where the variational problem foru⁰_Ω:

∀v ∈H0¹(Ω), Z

Ω

∇u⁰Ω(θ)· ∇v dx = Z

∂Ω

∂uΩ

∂n vθ·n ds.

was used in the last line, with test functionv =pΩ.

∀θ∈W^1,∞(R^d,R^d), J⁰(Ω)(θ) = Z

∂Ω

j(uΩ)−∂uΩ

∂n

∂pΩ

∂n

θ·n ds,

Mathematically speaking, it is therigorousway to assess the differentiability of shape functionals.

Several techniques presented above (in particular the adjoint technique) exist in much more general frameworks than shape optimization, and pertain to the framework ofoptimal control theory.

This way of obtaining shape derivatives is very involved in terms of calculations.

In practice, a formal method, which is much simpler, allows to calculate shape derivatives :C´ea’s method.

Contents

1 Several examples of shape optimization problems

2 Tools for shape optimization Generalities

The perimeter functional in shape optimization Existence results in shape optimization Derivation with respect to the domain

Shape derivatives using Eulerian and material derivatives : the rigorous ‘difficult’ way A formal, easier way to compute shape derivatives : C´ea’s method

3 Numerics and applications to Fluid Mechanics Reminders in Fluid Mechanics

Shape optimization in Fluid Mechanics Numerical aspects

The philosophy ofC´ea’s methodcomes from optimization theory :

Write the problem of minimizing J(Ω)as that of searching for the saddle points of a Lagrangian functional :

L(Ω,u,p) = Z

Ω

j(u)dx

| {z }

Objective function at stake

+ Z

Ω

(−∆u−f)p dx

| {z }

u=uΩ is enforced as a constraint by penalization with the Lagrange multiplierp

,

where the variables Ω,u,p areindependent.

This method isformal: in particular, it assumes that we already know that Ω7→uΩis differentiable.

C´ea’s method : the Neumann case (I)

Consider the followingLagrangian functional: L(Ω,v,q) =

Z

Ω

j(v)dx

| {z }

Objective function whereuΩ is replaced byv

+ Z

Ω

∇v· ∇q dx− Z

Ω

fq dx

| {z }

Penalization of the ’constraint’R v=uΩ : Ω (−∆v−f)q dx=0

,

which is defined for any shape Ω∈ Uad, and for anyv,q∈H¹(R^d),so that the variables Ω,v andq are independent.

One observes that, evaluatingLwithv=uΩ, it comes :

∀q∈H¹(R^d), L(Ω,uΩ,q) = Z

Ω

j(uΩ)dx=J(Ω).

For a fixed shape Ω, we search for thesaddle points(u,p)∈R^d×R^d ofL(Ω,·,·). The first-order necessary conditions read :

• ∀q∈H¹(R^d), ∂L

∂q(Ω,u,p)(q) = Z

Ω

∇u· ∇q dx− Z

Ω

fq dx= 0.

• ∀v ∈H¹(R^d), ∂L

∂v(Ω,u,p)(v) = Z

Ω

j⁰(u)·v dx+ Z

Ω

∇v· ∇p dx= 0.

C´ea’s method : the Neumann case (III)

Step 1 :Identification of u :

∀q∈H¹(R^d), Z

Ω

∇u· ∇q dx− Z

Ω

fq dx= 0.

Takingqas anyC^∞functionψwith compact support in Ω yields :

∀ψ∈ Cc^∞(Ω), Z

Ω

∇u· ∇ψdx− Z

Ω

fψdx= 0⇒ −∆u=f in Ω .

Now takingqas aC^∞ functionψand using Green’s formula :

∀ψ∈ Cc^∞(R^d), Z

∂Ω

∂u

∂nψds= 0⇒ ^∂u_∂n = 0 on∂Ω .

Conclusion :u=uΩ.

Step 2 :Identification of p :

∀v ∈H¹(R^d), Z

Ω

j⁰(u)v+ Z

Ω

∇v· ∇p dx= 0.

Takingv as anyC^∞functionψwith compact support in Ω yields :

∀ψ∈ Cc^∞(Ω), Z

Ω

∇p· ∇ψdx+ Z

Ω

j⁰(u)ψdx= 0

⇒ −∆u=−j⁰(uΩ) in Ω . Now takingv as aC^∞ functionψand using Green’s formula :

∀ψ∈ C^∞(R^d), Z

∂Ω

∂p

∂nϕds = 0⇒ ^∂p_∂n= 0 on∂Ω .

Conclusion :p=pΩ, solution to

−∆p=−j⁰(uΩ) inΩ

∂p

∂n = 0 on∂Ω

C´ea’s method : the Neumann case (V)

Step 3 :Calculation of the shape derivative J⁰(Ω)(θ): We go back to the fact that :

∀q∈H¹(R^d), L(Ω,uΩ,q) = Z

Ω

j(uΩ)dx.

Differentiating with respect to Ω yields :

∀θ∈W^1,∞(R^d,R^d), J⁰(Ω)(θ) = ∂L

∂Ω(Ω,uΩ,q)(θ) +∂L

∂v(Ω,uΩ,q)(u_Ω⁰(θ)), whereu⁰_Ω(θ) is theEulerian derivativeof Ω7→uΩ(assumed to exist).

Now, choosingq=pΩproduces, since ^∂L_∂v(Ω,uΩ,pΩ) = 0 :

J⁰(Ω)(θ) = ^∂L_∂Ω(Ω,uΩ,pΩ)(θ).

This last(partial)derivative amounts to the shape derivative of a functional of the form : Ω7→

Z

Ω

f(x)dx, wheref is afixedfunction.

Using the derivative of Ω7→R

Ωf, we end up with :

∀θ∈W^1,∞(R^d,R^d), J⁰(Ω)(θ) =

Z

∂Ω

(j(uΩ) +∇uΩ· ∇pΩ−fpΩ)θ·n ds.