theoretical and numerical aspects for incompressible ...frey/cours/upmc/nm... · sethian j.a.,level...

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

Pascal Frey and Yannick Privat(Slides by Charles Dapogny, Pascal Frey and Yannick Privat)

CNRS & Univ. Paris 6

feb. 2015

Y. Privat (CNRS & Univ. Paris 6) Master 2 - UPMC (2015) feb. 2015 1 / 81

Outlines of the lectures

Session 1 : tools for shapeoptimization in Fluid Mechanicsand existence of optimal shapes

Session 2 : Existence andoptimality condition (shapederivative)

Session 3 : Application in FluidMechanics and algorithms

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

Joining me : [email protected] the slides : http ://www.ann.jussieu.fr/frey/nm491.html


Outlines of the lesson

1 Several examples of shape optimization problems

2 Tools for shape optimizationGeneralitiesThe perimeter functional in shape optimizationExistence results in shape optimizationDerivation with respect to the domainShape derivatives using Eulerian and material derivatives : the rigorous ‘difficult’ wayA formal, easier way to compute shape derivatives : Cea’s method

3 Numerics and applications to Fluid MechanicsReminders in Fluid MechanicsShape optimization in Fluid MechanicsNumerical aspects


Several examples of shape optimization problems

Isoperimetric problems

The problem of Dido

Dido was the legendary founder of Carthage (Tunisia). When she arrived in814 B.C. on the coast of Tunisia, she asked for a piece of land. Her requestwas satisfied provided that the land could be encompassed by an ox-hide.With a remarkable mathematical intuition, she cut the ox-hide into a longthin strip and used it to encircle the land. This land became Carthage andDido became the Queen.

Question : What is the closedcurve which has the maximumarea for a given perimeter ?




Mathematically, this problem is very close to the standard isoperimetric one :

Find Ω ⊂ R2 solution of the problemmaximize area(Ω)such that Per(Ω) = 4 km.

Back to the problem of Dido. A naive formulation writes : find the plane curve enclosingwith the segment joining its extremity the subdomain having a maximal area. On otherwords, one has to solve for b > a ≥ 0,

supy∈E

∫ b

a

y(x) dx

where

E = y ∈ Y |∫ b

a

√1 + y ′2(x) dx = ` et y(a) = y(b) = 0

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




The proof

Zenodore (2nd century B.C.)proves the isoperimetricinequality in the particular casewhere Ω = polygon.

Until the 20th century : thisresult is conjectured butnot proved.

Steiner (Swiss mathematicianof the 19th century) publishes aproof, but . . . this proof iserroneous !

Weierstrass (German mathematician of the 19th century) concludes the proof, byusing modern tools of calculus of variations.

Generalization in R3 or RN only known since 1960 (geometric measure theory)



Aerodynamic optimization of airplaneAerodynamic shape optimization of transonic wings

The model

It requires the use of Navier-Stokesmodeling due to the strong nonlinearcoupling between airfoil shape, wavedrag, and viscous effects.

Criterion, unknown

Goal : reduce the drag (reaction ofthe flow on the wing ; itscomponent in the direction of flightis the drag proper and the rest isthe lift)

A few percent of drag optimizationmeans a great saving oncommercial airplanes ;

Unknown : the shape of the wing



Aerodynamic optimization of airplaneAerodynamic shape optimization of transonic wings

! For viscous drag the Navier-Stokes equations must be used : for a given shape S ofthe wing, it yields the velocity u and pressure p of the fluid at every point.

! For a wing with boundary S moving at constant speed u∞, the force acting on the

wing is ~F =

∫S

(µ(∇u + [∇u]>)− 2

3∇ · u

)n ds︸︷︷︸

viscous drag/lift = viscous force

−∫Spn ds, where n is the normal

to S pointing outside the domain occupied by the fluid, and µ the viscosity of thefluid.

! Shape optimization problem. The drag is the component of ~F parallel to the velocityat infinity. The optimal design problem writes

infS∈Oad

J(S) where J(Ω) = ~F · u∞ .

Choice of Oad (admissible set of shapes) ?

S is an open connected bounded subset of Rd (d = 2 or d = 3)A geometrical constraint such as the volume being greater than a given value in orderto avoid that the wing collapses (|S| ≥ S0)An aerodynamic constraint : the lift must be greater than a given value. Else, the wingwill not fly.



Optimal shape of a duct

Problem and modeling

Consider a fluid of viscosity µ flowing inside acannula-shaped pipe/duct. For instance, we lookfor the optimal shape of a pipeline.

The optimal design problem writes infΩ∈Oad

J(Ω) where J(Ω) = 2µ

∫Ω

|ε(u)|2 dx ,

with

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

Oad is the set of admissible shapes, for instance, E (inlet) and S (outlet) are fixedand we look for the lateral boundary such that Ω open connected subset of Rd with|Ω| = V0 (given).


Tools for shape optimization Generalities

Contents






Mathematical framework

A shape optimization problem writes as the minimization of a cost (or objective) functionJ of the domain Ω :

infΩ∈Oad

J(Ω),

where Oad is a set of admissible shapes (e.g. that satisfy constraints).

In most mechanical or physical applications, the relevant objective functions J(Ω) dependon Ω via a state uΩ, which arises as the solution to a PDE posed on Ω (e.g. the linearelasticity system, or Stokes equations).



Various settings for shape optimization (I)

I. Parametric optimization

The considered shapes are described by means of a set of physical parameterspii=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-section S is parame-

trized by its thickness function h : Ω→ R.



Various settings for shape optimization (II)

! The parameters describing shapes are the only optimization variables, and the shapeoptimization problem rewrites :

minpi∈Pad

J(p1, ..., pN),

where Pad is a set of admissible parameters.

! Parametric shape optimization is eased by the fact that it is straightforward toaccount for variations of a shape pii=1,...,N :

pii=1,...,N → pi + δpii=1,...,N .

! However, the variety of possible designs is severely restricted, and the use of such amethod 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) ofthe considered shapes is fixed.

! The boundary ∂Ω of the shapes Ω itself is theoptimization variable.

! Geometric optimization allows more freedomthan parametric optimization, since no a prioriknowledge of the relevant regions of shapes toact on is required.

@

Optimization of a shape by performing

‘free’ perturbations of its boundary.



Various settings for shape optimization (IV)

III. Topology optimization

In some applications, the suitable topology of shapesis unknown, and also subject to optimization.

In this context, it is often preferred not to describethe boundaries of shapes, but to resort to differentrepresentations which allow for a more naturalaccount of topological changes.

For instance : Describing shapes Ω ascharacteristic functions χΩ : 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 :A physical model, most often based on PDE (e.g. the linear elasticity equations,Stokes system, etc...) for describing the mechanical behavior of shapes,A description of shapes and their variations (e.g. as sets of parameters, densityfunctions, etc...),A numerical description of 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 optimizationshare a lot of common features.

! We are going to focus on geometric shape optimization methods.



Disclaimer

Disclaimer

I This course is very introductory, and by no means exhaustive, as well for theoreticalas for numerical purposes.

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



Bibliography/References

Allaire G., Conception optimale de structures, Mathematiques et Applications 58,Springer, (2006)

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

Henrot A., and Pierre M., Variation et optimisation de formes, une analysegeometrique, Springer, (2005)

Mohammadi B. and Pironneau O., Applied shape optimization for fluids, OxfordUniversity 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 inComputational Geometry, Fluid Mechanics, Computer Vision, and Materials Science,Cambridge University Press, (1999).

and many others. . .


Tools for shape optimization The perimeter functional in shape optimization

Contents






General definition of perimeter

Definition : Perimeter in the sense of De Giorgi

Let Ω be a (Lebesgue) mesurable set in Rd . The perimeter of Ω is defined by

Per(Ω) = sup∫

Ω

div(ϕ) dx | ϕ ∈ D(Rd ,Rd), ‖ϕ‖∞ ≤ 1,

where D(Rd ,Rd) is the set of C∞ functions ϕ : Rd → Rd having a compact support.

Definition introduced by Ennio De Giorgi (1928-1996)

Note that, for a function ϕ ∈ D(Rd ,Rd), there holds∫Ω

div(ϕ) dx =

∫Rd

div(ϕ)χΩ(x) dx = −〈∇χΩ, ϕ〉D′,D,

where χΩ denotes the characteristic function of Ω, in other words

χΩ(x) =

1 if x ∈ Ω0 else.




Reminder on the notion of “regular set”

1 An open set Ω of Rd is said to have a Lipschitz boundary if, for all x0 ∈ ∂Ω, thereexist a cylinder K = K ′×]− a, a[ in a local cartesian basis with origin x0 = 0, withK ′ an open ball in Rd−1 with radius r , and a Lipschitz function ϕ : K ′ →]− a, a[such that ϕ(0) = 0 and

Ω ∩ K = (x ′, y) ∈ K | y > ϕ(x ′) et ∂Ω ∩ K = (x ′, ϕ(x ′)) | x ′ ∈ K ′.

2 An open set is said of class C1 by replacing the word ”Lipschitz” by C1 in thedefinition above.

Proposition (Link with the usual definition of perimeter for regular sets)

Let Ω, be a bounded open set of class C1. Then, Per(Ω) =∫∂Ω

dσ, where dσ denotes thesurface element on ∂Ω.




Proposition



Sketch of the proof : let ϕ ∈ D(Rd ,Rd) such that ‖ϕ‖∞ ≤ 1. According to Green’sformula, one has∫

Ω

div(ϕ) dx =d∑

i=1

∫Ω

∂ϕi

∂xidx =

∫∂Ω

ϕini dσ =

∫∂Ω

ϕ · ν dσ,

where ν is the outward unit normal vector. Using the triangle inequality, one has for everysmooth function ϕ,∫

Ω

div(ϕ) dx ≤∫∂Ω

|ϕ||ν| dσ ≤ ‖ϕ‖∞∫∂Ω

dσ ≤∫∂Ω

dσ.

Passing to the sup, one gets Per(Ω) ≤∫∂Ω

dσ.




Proposition



Proof (converse sense) : One would like to choose ϕ = ν on ∂Ω.Idea : find a sequence of functions (ϕn)n∈N ∈ D(Rd ,Rd)N to approximate ν. Introduce N ,a continuous extension of the normal vector ν. One regularizes N by defining ϕn = N ∗ρn,where (ρn)n∈N is a mollifier.One thus construct a sequence of functions in D(Rd ,Rd) converging uniformly to N ands.t. ∫

Ω

divϕn dx =

∫∂Ω

ϕn · ν dσ −−−−→n→+∞

∫∂Ω

ν · ν dσ =

∫∂Ω

dσ.




Reminder on Radon measures : recall that if f ∈ L1(Rd ,Rd), then by duality

‖f ‖L1 =

∫Rd

|f (x)| dx = sup∫Rd

f (x) · ϕ(x) dx , ϕ ∈ D(Rd ,Rd) and ‖ϕ‖∞ ≤ 1.

More generally, this formula makes sense for every Radon measure µ having a finitemass (i.e. every linear continuous form on the space C0

0 (Rd) of continuous functionshaving a compact support in Rd)

Moreover, a Radon measure µ has a finite mass on Rd iff the quantity

‖µ‖1 = sup〈µ, ϕ〉D′,D, ϕ ∈ D(Rd ,Rd) et ‖ϕ‖∞ ≤ 1

is finite.

Recall that

Per(Ω) = sup−〈∇χΩ, ϕ〉D′,Dϕ ∈ D(Rd ,Rd), ‖ϕ‖∞ ≤ 1

Proposition

Let Ω be a measurable subset of Rd . Then, the quantity Per(Ω) is finite iff ∇χΩ is a Radonmeasure having a finite mass and in that case, Per(Ω) = ‖∇χΩ‖1.




Theorem (compactness properties of the perimeter)

Let (Ωn)n∈N be a sequence of measurable sets in Rd . Let us assume the existence of C > 0s.t.

∀n ∈ N, |Ωn|+ Per(Ωn) ≤ C .

Then, there exist a measurable set Ω ⊂ Rd and a subsequence (Ωnk )k∈N s.t.

χΩnk→ χΩ dans L1

loc(Rd) et 〈∇χΩnk, ϕ〉 → 〈∇χΩ, ϕ〉, ∀ϕ ∈ C0

0 (Rd).

Moreover, if all the elements of the sequence (Ωn)n∈N belong to an open set D having afinite measure, then the sequence (χΩnk

)k∈N converges to χΩ in L1(D).

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

! The proof of this theorem rests upon the compactness of the injectionBV (Rd) → L1

loc(Rd).




Definition : convergence in the sense of characteristic functions.

Let (Ωn)n∈N and Ω denote respectively a sequence of measurable sets and a measurableset of Rd . The sequence (Ωn)n∈N is said to converge in the sense of characteristicfunctions if χΩn → χΩ in L1

loc(Rd).

According to the previous theorem, a sequence of domains contained in a boundedopen set D and whose perimeter is uniformly bounded converges (up to asubsequence) in the sense of characteristic functions.

Let (Ωn)n∈N be a sequence of measurable subsets of Rd . Without additionalassumption, one cannot conclude that (χΩn )n∈N converges up to a subsequence toχΩ. Nevertheless, since L∞(Rd ; 0, 1) is the topological dual of the spaceL1(Rd ; 0, 1), there exist a subsequence (χΩnk

)k∈N converging weakly-? in L∞ to a

function a ∈ L∞(Rd , [0, 1]) (Banach-Alaoglu theorem).

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

⋃nk=1

(kπn+1− π

4n, kπn+1

+ π4n

), for every n ∈ N∗.

One can prove (exercise) : |ωn| = π2

, and the sequence (χωn )n∈N converges weakly-∗in L∞ to the constant function a(·) = 1

2.


Tools for shape optimization Existence results in shape optimization

Contents






Existence of solutions in shape optimization

General issues

Let us consider the general optimal design problem

infΩ∈Oad

J(Ω) with Oad ⊂ A(D) = Ω ⊂ D, Ω open,

where D is an open set of Rd .

Study plan for existence issues

1 is the quantity infΩ∈Oad J(Ω) finite ?In that case, introduce a minimizing sequence (Ωn)n∈N of elements of Oad.Moreover, the sequence (J(Ωn))n∈N is bounded.

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

topology ?→ that the functional J be lower semi-continuous for this topology, in other words so that

Ωn → Ω ⇒ J(Ω) ≤ lim infn→+∞

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

Let D a bounded connected open set and Oad be a subset of

A(D) = Ω open, Ω ⊂ D.

We consider the shape optimization problem

infΩ∈Oad

J(Ω) where J(Ω) =

∫Ω

(uΩ(x)− u0(x))2 dx ,

with u0 ∈ L2(D) given, and uΩ the unique solution of the P.D.E.−∆uΩ = f x ∈ ΩuΩ = 0 x ∈ ∂Ω,

with f ∈ H−1(D).To obtain the existence of a solution, we will have, in addition to the convergence of theminimizing sequence (Ωn)n∈N in a sense to make precise, to obtain the convergence of(uΩn )n∈N in L2(D) for example.




Definition (Hausdorff complement topology)

Let x ∈ Rd and E a closed subset of Rd . We denote

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

Hausdorff distance between two closed sets : for A, B, two closed sets of Rd ,

dH(A,B) = maxsupx∈A

dB(x), supx∈B

dA(x).

Convergence of open sets in the sense of Hausdorff (Hausdorff complementarytopology) : for A and B in A(D),

dHc (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 thesense of Hausdorff if dHc (Ωn,Ω)→ 0 as n→ +∞.




Remarks on Hausdorff convergence

Very important property

For every A,B closed subsets of D ⊂ Rd ,

dH(A,B) = ‖dA − dB‖C0(D) := supx∈D|dA(x)− dB(x)| .

Proof : there holds

‖dA − dB‖C0(D) ≥ max‖dA − dB‖C0(A), ‖dA − dB‖C0(B) ≥ dH(A,B).

Let us show the converse inequality. Let x ∈ B and y ∈ A. By compactness, there exist xB ∈ Bs.t. dB(x) = |x − xB |. Hence,

dA(x)− dB(x) ≤ |y − x | − |x − xB | ≤ |y − xB | ≤ dA(xB) ≤ supx∈B

dA(x).

By inverting the roles played by A and B, we infer the desired inequality.




Back to the Hausdorff complement topology

Proposition

1 Let A and B in D. One has dH(A,B) = 0 iff A = B.

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

Cd(D) = dΩ, with Ω 6= ∅ and Ω ⊂ D with dΩ : x 7→ d(x ,Ω).

Then, Cd(D) is a compact subspace of C0(D).

! It is enough to restrict the set of admissible shapes to classes of open sets. Indeed,

dΩc , Ω ⊂ D and Ω 6= ∅ = dΩc , Ω open set included in D.

To show this equality, consider Ω ⊂ D s.t. Ω 6= ∅.Consider the open set O = Ωc

c. Since Oc = Ωc , there holds dOc = dΩc = dΩc .

Therefore, the proposition above must be interpreted in the following sense : if(Ωn)n∈N is a sequence of A(D) converging to Ω in the sense of Hausdorff, then thelimits of the sequence (Ωn)n∈N belong to the same equivalence class and there existsΩ ∈ A(D) s.t. (Ωn)n∈N converges to Ω.




Proposition (compactness and l.s.c. for the topology Hc)

1 The set A(D) endowed with the metric dHc is compact.

2 Let (Ωn)n∈N, a sequence of elements in A(D) that converges in the sense ofHausdorff to Ω. For every compact subset K ⊂ Ω, there exists an integer N(K) ∈ Nsuch that K ⊂ Ωn for every n ≥ N(K).

3 The Lebesgue measure is lower semi-continuous for the topology dHc .

Examples (exercise)

Consider an open set D and (xn)n∈N, a sequence of dense points in D. SetΩn = D\xk1≤k≤n. Then, (Ωn)n∈N converges in the sense of Hausdorff to ∅.Ωn =]0, 1[\ 1

nn∈N converges in the sense of Hausdorff to ]0, 1[.



The uniform cone property

Definition

Let y , a point of Rd , ξ, a unit vector and ε > 0.

1 Introduce the cone C(y , ξ, ε) with apex y , direction ξ and dimension ε, defined by

C(y , ξ, ε) = z ∈ Rd | 〈z − y , ξ〉Rd ≥ cos ε‖z − y‖Rd and 0 < ‖z − y‖Rd < ε,

where 〈·, ·〉Rd is the euclidean scalar product of Rd and ‖ · ‖Rd , the associatedeuclidean norm.

2 An open set Ω verifies the ε-cone property if for every x ∈ Ω, there exists ξx , a unitvector such that :

∀y ∈ Ω ∩ B(x , ε), C(y , ξx , ε) ⊂ Ω,

where B(x , ε) denotes the open ball with center x and radius ε.

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

! Theorem : an open bounded set Ω satisfies the ε-cone property iff it has a Lipschitzboundary.




C(y, ξx, ε)

y x

ε

ξx

Illustration of the ε-cone property

Left : set verifying the ε-cone property,Right : set that does notverify the ε-cone property




Compactness property of ε-cone property

Proposition

Let ε > 0 and (Ωn)n∈N, a sequence of open sets contained in a bounded open set Dsatisfying the ε-cone property for some ε > 0.Then, there exist an open set Ω and a subsequence (Ωnk )k∈N converging to Ω in the senseof Hausdorff. Moreover,

Ω has the ε-cone property,

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

one may assume that (Ωnk )k∈N also converges to Ω in the sense of characteristicfunctions.



Existence issues : two examples

Example 1 : an isoperimetric like problem

infΩ∈OV0

Per(Ω) avec OV0 = Ω ⊂ D |∫

Ω

f (x) dx = V0 ,

with D, a bounded open set of Rd , f ∈ L1loc(Rd) and V0 > 0 given.

Theorem

The problem above has (at least) a solution.




Theorem


Proof : let (Ωn)n∈N be a minimizing sequence in OV0 . Then, there exists c > 0 s.t.Per(Ωn) ≤ c for all n ∈ N.Since Ωn ⊂ D for all n ∈ N, one infers by compactness of sets having a bounded perimeterthat : ∃(Ωnk )k∈N and a measurable set Ω s.t. χΩnk

→ χΩ in L1loc(Rd). Therefore,∫

Ωnk

f (x) dx =

∫Rd

χΩnk(x)f (x) dx −−−−→

n→+∞

∫Ω

f (x) dx .

Let ϕ ∈ D(Rd ,Rd). Hence,∫Ω

div(ϕ) dx = limn→+∞

∫Ωnk

div(ϕ) dx ≤ lim infk→+∞

supϕ∈D(Rd ,Rd ),‖ϕ‖∞≤1

∫Ωnk

div(ϕ) dx

= lim infk→+∞

Per(Ωnk ).

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




Example 2 : back to the toy problem

Let ε > 0 and D be a bounded open set of Rd .

infΩ∈Oad,ε

J(Ω) where J(Ω) =

∫Ω

(uΩ(x)− u0(x))2 dx ,

whereOad,ε = Ω ∈ A(D) | |Ω| ≤ 1 and Ω satisfies the ε-cone property,

with u0 ∈ L2(D) given, and uΩ the unique solution of the P.D.E.−∆uΩ = f x ∈ ΩuΩ = 0 x ∈ ∂Ω,

with f ∈ H−1(D).

Theorem





Theorem


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

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

Ω satisfies the ε-cone property ;

|Ω| ≤ lim infk→+∞ |Ωnk | ≤ 1 (the measure is l.s.c. for the Hausdorff topology) ;

Multiply −∆uΩnk= f by uΩnk

and integrate by parts. It yields

‖uΩnk‖H1

0 (D) ≤ 〈f , uΩnk〉H−1,H1

0≤ ‖f ‖H−1(D)‖uΩnk

‖H10 (D).

Thus, (uΩnk)k is bounded in H1(D) and according to Rellich-Kondratov theorem, it

converges to u∗ ∈ H10 (D) weakly in H1 and strongly in L2.




Theorem


Proof :

(uΩnk)k∈N is bounded in H1(D) and according to Rellich-Kondratov theorem, it

converges to u∗ ∈ H10 (D) weakly in H1(D) and strongly in L2(D).

Passing to the limit in the variational formulation

∀ϕ ∈ H10 (Ω),

∫D

∇uΩnk· ∇ϕ dx = 〈f , ϕ〉H−1,H1

0

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

Writing uΩnk(x)(χΩnk

(x)− χD(x)) = 0 for almost every x ∈ D. Uand passing to thelimit yields u∗ = 0 almost everywhere in D\Ω∗.Consequently, u∗ ∈ H1

0 (Ω∗) and then u∗ solves the P.D.E.−∆u∗ = f x ∈ Ωu∗ = 0 x ∈ ∂Ω,

Conclusion : inf J = limk→+∞ J(Ωnk ) =∫

Ω(u∗ − u0)2 dx whence the existence.


Tools for shape optimization Derivation with respect to the domain

Contents






Differentiation with respect to the domain : Hadamard’s method

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

Ω 7→ Ωθ := (I + θ)(Ω),

for ‘small’ vector fields θ ∈W 1,∞ (Rd ,Rd

).

•

•

•x

(x)

Lemma

For θ ∈W 1,∞(Rd ,Rd) such that ||θ||W 1,∞(Rd ,Rd )< 1, the application (I + θ) is aLipschitz diffeomorphism.



Differentiation with respect to the domain : Hadamard’s method

Definition

Given a smooth domain Ω, a scalar function Ω 7→ J(Ω) ∈ R is said to be shapedifferentiable at Ω if the function

W 1,∞(Rd ,Rd

)3 θ 7→ J(Ωθ)

is Frechet-differentiable at 0, i.e. the following expansion holds in the vicinity of 0 :

J(Ωθ) = J(Ω) + J ′(Ω)(θ) + o(||θ||W 1,∞(Rd ,Rd)

).

The linear mapping θ 7→ J ′(Ω)(θ) is the shape derivative of J at Ω.



Structure of shape derivatives (I)

Idea : The shape derivative J ′(Ω)(θ) of a ‘regular’ functional J(Ω) only depends on thenormal component θ · n of the vector field θ.

Figure: At first order, a tangential vector field θ, (i.e. θ · n = 0) only results in a convection ofthe shape Ω, and it is expected that J′(Ω)(θ) = 0.



Structure of shape derivatives (II)

Lemma

Let Ω be a domain of class C1. Assume that the application

C1,∞(Rd ,Rd) 3 θ 7→ J(Ωθ) ∈ R

is of class C1. Then, for any vector field θ ∈ C1,∞(Rd ,Rd) such that θ · n = 0 on ∂Ω, onehas : J ′(Ω)(θ) = 0.

Corollary

Under the same hypotheses, if θ1, θ2 ∈ C1,∞(Rd ,Rd) have the same normal component,i.e. θ1 · n = θ2 · n on ∂Ω, then :

J ′(Ω)(θ1) = J ′(Ω)(θ2).



Structure of shape derivatives (III)

Actually, the shape derivatives of ‘many’ integral objective functionals J(Ω) can be putunder the form :

J ′(Ω)(θ) =

∫∂Ω

vΩ (θ · n) ds,

where vΩ : ∂Ω→ R is a scalar field which depends on J and on the current shape Ω.

This structure lends itself to the calculation of a descent direction : letting θ = −tvΩn,for a small enough descent step t > 0 yields :

J(Ωtθ) = J(Ω)− t

∫∂Ω

v 2Ω ds + o(t) < J(Ω).



First examples of shape derivatives

Theorem

Let Ω ⊂ Rd be a bounded Lipschitz domain, and f ∈ W 1,1(Rd) be a fixed function.Consider the functional :

J(Ω) =

∫Ω

f (x) dx ;

then J is shape differentiable at Ω and its shape derivative is :

∀θ ∈W 1,∞(Rd ,Rd), J ′(Ω)(θ) =

∫∂Ω

f (θ · n) ds.




•

•

•

xx + (x)

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




Remarks :

This result is actually a particular case of the Transport (or Reynolds) theorem, usedto derive the equations of conservation from conservation principles.

It allows to calculate the shape derivative of the volume functionalVol(Ω) =

∫Ω

1 dx :

∀θ ∈W 1,∞(Rd ,Rd), Vol′(Ω)(θ) =

∫∂Ω

θ · n ds =

∫Ω

div(θ) dx .

In particular, if div(θ) = 0, the volume does not vary (at first order) when Ω isperturbed by θ.




Proof : The formula proceeds from a change of variables :

J(Ωθ) =

∫(I+θ)(Ω)

f (x)dx =

∫Ω

|det(I +∇θ)| f (I + θ) dx .

• The mapping θ 7→ det(I +∇θ) is Frechet differentiable at 0, and :

det(I +∇θ) = 1 + div(θ) + o(θ),o(θ)

||θ||W 1,∞(Rd ,Rd )−→θ→0

0.

• If f ∈W 1,1(Rd), θ 7→ f (I + θ) is also Frechet differentiable at 0 and :

f (I + θ) = f +∇f · θ + o(θ).

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




Theorem

Let Ω0 ⊂ Rd be a bounded, regular enough domain, and g ∈W 2,1(Rd) be a fixed function.Consider the functional :

J(Ω) =

∫∂Ω

g(x) ds;

then J is shape differentiable at Ω0 and its shape derivative is :

J ′(Ω)(θ) =

∫∂Ω

(∂g

∂n+ κg

)(θ · n) ds,

where κ stands for the mean curvature of ∂Ω.

Example : The shape derivative of the perimeter P(Ω) =∫∂Ω

1 ds is :

P ′(Ω)(θ) =

∫∂Ω

κ (θ · n) ds.



Towards more sophisticated examples

The examples of physical interest are those of PDE constrained shape optimization, i.e.one aims at minimizing functions which depend on Ω via the solution uΩ of a PDE posedon Ω, for instance (in most of the forthcoming examples) :

J(Ω) =

∫Ω

j(uΩ) dx +

∫∂Ω

k(uΩ) ds,

where uΩ is e.g. the solution to the linear elasticity system posed on Ω, and j , k are givenfunctions.

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



The framework

Henceforth, we rely on the model of the Laplace equation : the state uΩ is solutionto the system

−∆u = f in Ωu = 0 on ∂Ω (Dirichlet B.C)∂u∂n

= 0 on ∂Ω (Neumann B.C)

where∫

Ωf dx = 0 in the Neumann case.

The associated variational formulation reads :

∀v ∈ H10 (Ω)/H1(Ω),

∫Ω

∇u · ∇v dx −∫

Ω

fv dx = 0.

We aim at calculating the shape derivative of J(Ω) =∫

Ωj(uΩ) dx , where j : R→ R

is a ‘smooth enough’ function.


Tools for shape optimizationShape derivatives using Eulerian and material derivatives : the rigorous

‘difficult’ way

Contents






‘difficult’ way

Eulerian and Lagrangian derivatives (I)

The rigorous way to address this problem requires a notion of differentiation of functionsΩ 7→ uΩ, which to a domain Ω associate a function defined on Ω. One could think of twoways of doing so :

The Eulerian point of view : For a

fixed x ∈ Ω, u′Ω(θ)(x) is the derivativeof the application

θ 7→ uΩθ (x).

•x

u

u

The Lagrangian point of view : For

a fixed x ∈ Ω, uΩ(θ)(x) is the derivativeof the application

θ 7→ uΩθ ((I + θ)(x)).

•x

(x)

• x + (x)



‘difficult’ way

Eulerian and Lagrangian derivatives (II)

The Eulerian notion of shape derivative, however more intuitive, is more difficult todefine rigorously. In particular, differentiating the boundary conditions satsified by uΩ

is clumsy :even for θ ‘small’, uΩθ (x) may not make any sense if x ∈ ∂Ω !

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

The Eulerian derivative will be defined after the Lagrangian derivative, from theformal use of chain rule over the expression u(I+θ)(Ω) (I + θ) :

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



‘difficult’ way

Eulerian and Lagrangian derivatives (III)

Let Ω 7→ u(Ω) ∈ H1(Ω) be a function which to a domain, associates a function on thedomain.

Definition

The function u : Ω 7→ u(Ω) admits a material, or Lagrangian derivative u(Ω) at a givendomain Ω provided the transported function

W 1,∞(Rd ,Rd) 3 θ 7−→ u(θ) := u(Ωθ) (I + θ) ∈ H1(Ω),

which is defined in the neighborhood of 0 ∈W 1,∞(Rd ,Rd), is differentiable at θ = 0.



‘difficult’ way

Eulerian and Lagrangian derivatives (IV)

We are now in position to define the notion of Eulerian derivative.

Definition

The function u : Ω 7→ u(Ω) admits a Eulerian derivative u′(Ω)(θ) at a given domain Ω inthe direction θ ∈W 1,∞(Rd ,Rd) if it admits a material derivative u(Ω)(θ) at Ω, and∇u(Ω) · θ ∈ H1(Ω).One defines then :

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



‘difficult’ way

Eulerian and Lagrangian derivatives (V)

Proposition

Let Ω ⊂ Rd be a smooth bounded domain, and suppose that Ω 7→ u(Ω) has a Lagrangianderivative u(Ω) at Ω. If j : R → R is regular enough, the function J(Ω) =

∫Ωj(u(Ω)) dx

is then shape differentiable at Ω, and :

J ′(Ω)(θ) =

∫Ω

(u(Ω)(θ)j ′(u(Ω)) + j(u(Ω)) div(θ)

)dx .

for every θ ∈W 1,∞(Rd ,Rd).If u(Ω) has a Eulerian derivative u′(Ω) at Ω, one has the ‘chain rule’ :

J ′(Ω)(θ) =

∫∂Ω

j(u(Ω)) θ · n ds︸︷︷︸Derivative of Ω 7→

∫Ω j(uΩ)

with respect to its first argument

+

∫Ω

j ′(u(Ω))u′(Ω)(θ) dx︸︷︷︸Derivative of Ω 7→

∫Ω j(uΩ)

with respect to its second argument

.

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



‘difficult’ way

Eulerian and Lagrangian derivatives (VI)

Let us return to our problem of calculating the shape derivative of :

J(Ω) =

∫Ω

j(uΩ) dx , where

−∆u = f in Ωu = 0 on ∂Ω

.

The following result characterizes the Lagrangian derivative of Ω 7→ uΩ. Its proof can beadapted to many different PDE models :

Theorem

Let Ω ⊂ Rd be a smooth bounded domain. The application Ω 7→ uΩ ∈ H10 (Ω) admits a

Lagrangian derivative uΩ(θ), and for any θ ∈W 1,∞(Rd ,Rd), uΩ(θ) ∈ H10 (Ω) is the unique

solution to : −∆Y = −∆(θ · ∇uΩ) in Ω

Y = 0 on ∂Ω.



‘difficult’ way

Eulerian and Lagrangian derivatives (VII)

Idea of the proof : The variational problem satisfied by uΩθ is :

∀v ∈ H10 (Ωθ),

∫Ωθ

∇uΩθ · ∇v dx =

∫Ωθ

fv dx .

By a change of variables, the transported function u(θ) := uΩθ (I + θ) satisfies :

∀v ∈ H10 (Ω),

∫Ω

A(θ)∇u(θ) · ∇v dx =

∫Ωθ

|det(I +∇θ)|fv dx ,

where A(θ) := |det(I +∇θ)|∇θ∇θT .

This variational problem features a fixed domain and a fixed function space H10 (Ω), and

only the coefficients of the formulation depend on θ.



‘difficult’ way

Eulerian and Lagrangian derivatives (VIII)

The problem can now be written as an equation for u(θ) :

F(θ, u(θ)) = G(θ),

for appropriate definitions of the operators :F : W 1,∞(Rd ,Rd )× H1

0 (Ω)→ H−1(Ω),

G : W 1,∞(Rd ,Rd )→ H−1(Ω).

A use of the implicit function theorem provides the result.

In particular, the Lagrangian derivative uΩ(θ) satisfies :

∀v ∈ H10 (Ω),

∫Ω

∇uΩ(θ) · ∇v dx =

∫Ω

∇(θ · ∇uΩ) · ∇v dx .



‘difficult’ way

Eulerian and Lagrangian derivatives (IX)

Remark : The Eulerian derivative of uΩ can now be computed from its Lagrangianderivative. It satisfies :

−∆U = 0 in Ω

U = −(θ · n) ∂uΩ∂n

on ∂Ω,

or under variational form :

∀v ∈ H10 (Ω),

∫Ω

∇u′Ω(θ) · ∇v dx =

∫∂Ω

∂uΩ

∂nv θ · n ds.

Using this formula in combination with :

J ′(Ω)(θ) =

∫∂Ω

j(uΩ) θ · n ds +

∫Ω

j ′(uΩ)u′Ω(θ) dx

will allow to express J ′(Ω)(θ) as a completely explicit expression of θ : this is the adjointmethod from optimal control theory.

Goal of the adjoint method : write J ′(Ω)(θ) under the form

J ′(Ω)(θ) =

∫∂Ω

something× (θ · n) ds

where something does not depend on θ.



‘difficult’ way

Eulerian and Lagrangian derivatives (X) : the adjoint method

Idea : ‘lift up’ the term of J ′(Ω)(θ) which features the Eulerian derivative of uΩ

by introducing an adequate auxiliary problem.

Let pΩ ∈ H10 (Ω) be defined as the solution to the problem :

−∆p = −j ′(uΩ) in Ωp = 0 on ∂Ω

.

The variational formulation for pΩ is :

∀v ∈ H10 (Ω),

∫Ω

∇pΩ · ∇v dx = −∫

Ω

j ′(uΩ) v dx ,

... to be compared with :

J ′(Ω)(θ) =

∫∂Ω

j(uΩ) θ · n ds +

∫Ω

j ′(uΩ)u′Ω(θ) dx .



‘difficult’ way

Eulerian and Lagrangian derivatives (XI) : the adjoint method

Thus, J ′(Ω)(θ) =

∫∂Ω

j(uΩ) θ · n ds +

∫Ω

j ′(uΩ)u′Ω(θ) dx

=

∫∂Ω

j(uΩ) θ · n ds −∫

Ω

∇pΩ · ∇u′Ω(θ) dx

=

∫∂Ω

j(uΩ) θ · n ds −∫∂Ω

∂pΩ

∂n

∂uΩ

∂nθ · n dx

,

where the variational problem for u′Ω :

∀v ∈ H10 (Ω),

∫Ω

∇u′Ω(θ) · ∇v dx =

∫∂Ω

∂uΩ

∂nv θ · n ds.

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

∀θ ∈W 1,∞(Rd ,Rd), J ′(Ω)(θ) =

∫∂Ω

(j(uΩ)− ∂uΩ

∂n

∂pΩ

∂n

)θ · n ds,



‘difficult’ way

Eulerian and Lagrangian derivatives : summary

Mathematically speaking, it is the rigorous way to assess the differentiability ofshape functionals.

Several techniques presented above (in particular the adjoint technique) exist inmuch more general frameworks than shape optimization, and pertain to theframework of optimal 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 shapederivatives : Cea’s method.


Tools for shape optimization A formal, easier way to compute shape derivatives : Cea’s method

Contents






Cea’s method

The philosophy of Cea’s method comes from optimization theory :

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

L(Ω, u, p) =

∫Ω

j(u) dx︸︷︷︸Objective function at stake

+

∫Ω

(−∆u − f )p dx︸︷︷︸u=uΩ is enforced as a constraint

by penalization with the Lagrange multiplier p

,

where the variables Ω, u, p are independent.

This method is formal : in particular, it assumes that we already know that Ω 7→ uΩ isdifferentiable.



Cea’s method : the Neumann case (I)

Consider the following Lagrangian functional :

L(Ω, v , q) =

∫Ω

j(v) dx︸︷︷︸Objective function

where uΩ is replaced by v

+

∫Ω

∇v · ∇q dx −∫

Ω

fq dx︸︷︷︸Penalization of the ’constraint’ v=uΩ:∫

Ω (−∆v−f )q dx=0

,

which is defined for any shape Ω ∈ Uad , and for any v , q ∈ H1(Rd), so that the variablesΩ, v and q are independent.

One observes that, evaluating L with v = uΩ, it comes :

∀q ∈ H1(Rd), L(Ω, uΩ, q) =

∫Ω

j(uΩ) dx = J(Ω).



Cea’s method : the Neumann case (II)

For a fixed shape Ω, we search for the saddle points (u, p) ∈ Rd × Rd of L(Ω, ·, ·). Thefirst-order necessary conditions read :

• ∀q ∈ H1(Rd),∂L∂q

(Ω, u, p)(q) =

∫Ω

∇u · ∇q dx −∫

Ω

fq dx = 0.

• ∀v ∈ H1(Rd),∂L∂v

(Ω, u, p)(v) =

∫Ω

j ′(u) · v dx +

∫Ω

∇v · ∇p dx = 0.



Cea’s method : the Neumann case (III)

Step 1 : Identification of u :

∀q ∈ H1(Rd),

∫Ω

∇u · ∇q dx −∫

Ω

fq dx = 0.

Taking q as any C∞ function ψ with compact support in Ω yields :

∀ψ ∈ C∞c (Ω),

∫Ω

∇u · ∇ψ dx −∫

Ω

f ψ dx = 0 ⇒ −∆u = f in Ω .

Now taking q as a C∞ function ψ and using Green’s formula :

∀ψ ∈ C∞c (Rd),

∫∂Ω

∂u

∂nψ ds = 0 ⇒ ∂u

∂n= 0 on ∂Ω .

Conclusion : u = uΩ.



Cea’s method : the Neumann case (IV)

Step 2 : Identification of p :

∀v ∈ H1(Rd),

∫Ω

j ′(u)v +

∫Ω

∇v · ∇p dx = 0.

Taking v as any C∞ function ψ with compact support in Ω yields :

∀ψ ∈ C∞c (Ω),

∫Ω

∇p · ∇ψ dx +

∫Ω

j ′(u)ψ dx = 0

⇒ −∆u = −j ′(uΩ) in Ω .

Now taking v as a C∞ function ψ and using Green’s formula :

∀ψ ∈ C∞(Rd),

∫∂Ω

∂p

∂nϕ ds = 0 ⇒ ∂p

∂n= 0 on ∂Ω .

Conclusion : p = pΩ, solution to

−∆p = −j ′(uΩ) in Ω

∂p∂n

= 0 on ∂Ω



Cea’s method : the Neumann case (V)

Step 3 : Calculation of the shape derivative J ′(Ω)(θ) :

We go back to the fact that :

∀q ∈ H1(Rd), L(Ω, uΩ, q) =

∫Ω

j(uΩ) dx .

Differentiating with respect to Ω yields :

∀θ ∈W 1,∞(Rd ,Rd), J ′(Ω)(θ) =∂L∂Ω

(Ω, uΩ, q)(θ) +∂L∂v

(Ω, uΩ, q)(u′Ω(θ)),

where u′Ω(θ) is the Eulerian derivative of Ω 7→ uΩ (assumed to exist).

Now, choosing q = pΩ produces, since ∂L∂v

(Ω, uΩ, pΩ) = 0 :

J ′(Ω)(θ) = ∂L∂Ω

(Ω, uΩ, pΩ)(θ).



Cea’s method : the Neumann case (VI)

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

Ω 7→∫

Ω

f (x) dx ,

where f is a fixed function.

Using the derivative of Ω 7→∫

Ωf , we end up with :

∀θ ∈W 1,∞(Rd ,Rd),

J ′(Ω)(θ) =

∫∂Ω

(j(uΩ) +∇uΩ · ∇pΩ − fpΩ) θ · n ds.



Cea’s method : the Dirichlet case (I)

We now consider the problem of derivating :

J(Ω) =

∫Ω

j(uΩ) dx , where

−∆u = f in Ωu = 0 on ∂Ω

.

Warning : When the state uΩ satisfies essential boundary conditions, i.e. boundary

conditions that are tied to the definition space of functions (here, H10 (Ω)), an

additional difficulty arises.

We can no longer use the Lagrangian

L(Ω, v , q) =

∫Ω

j(v) dx +

∫Ω

∇v · ∇q dx −∫

Ω

fv dx ,

since it would have to be defined for v , q ∈ H10 (Ω).

In this case, the variables Ω, v , q would not be independent.



Cea’s method : the Dirichlet case (II)

Solution : Add an extra variable µ ∈ H1(Rd) to the Lagrangian to penalize the

boundary condition : for all v , q, µ ∈ H1(Rd) ;

L(Ω, v , q, µ) =

∫Ω

j(v) dx︸︷︷︸Objective function

where uΩ is replaced by v

+

∫Ω

(−∆v − f )q dx︸︷︷︸penalization of the

‘constraint’−∆v=f

+

∫∂Ω

µv ds︸︷︷︸penalization of the

‘constraint’ v=0 on ∂Ω

.

By Green’s formula, L rewrites :

L(Ω, v , q, µ) =

∫Ω

j(v) dx +

∫Ω

∇v · ∇q dx −∫

Ω

fq dx +

∫∂Ω

(µv − ∂v

∂nq

)ds.

Of course, evaluating L with v = uΩ, it comes :

∀q, µ ∈ H1(Rd), L(Ω, uΩ, q) =

∫Ω

j(uΩ) dx .



Cea’s method : the Dirichlet case (III)

For a fixed shape Ω, we look for the saddle points (u, p, ν) ∈ (H1(Rd))3 of the functionalL(Ω, ·, ·, ·). The first-order necessary conditions are :

• ∀q ∈ H1(Rd),∂L∂q

(Ω, u, p, λ)(q) = ∫Ω

∇u · ∇q dx −∫

Ω

fq dx +

∫∂Ω

∂u

∂nq ds = 0.

• ∀v ∈ H1(Rd),∂L∂v

(Ω, u, p, λ)(v) =∫Ω

j ′(u) · v dx +

∫Ω

∇v · ∇p dx +

∫∂Ω

(λv − ∂v

∂np

)ds = 0.

• ∀µ ∈ H1(Rd),∂L∂µ

(Ω, u, p, λ)(µ) =

∫∂Ω

µu ds = 0.



Cea’s method : the Dirichlet case (IV)

Step 1 : Identification of u :

∀q ∈ H1(Rd),

∫Ω

∇u · ∇q dx −∫

Ω

fq dx +

∫∂Ω

∂u

∂nq ds = 0.


∀ψ ∈ C∞c (Ω),

∫Ω

∇u · ∇ψ dx =

∫Ω

f ψ dx ⇒ −∆u = f in Ω .

Using ∂L∂µ

(Ω, u, pλ)(µ) = 0 for any µ = ψ ∈ C∞c (Rd) yields :

∀ψ ∈ C∞c (Rd),

∫∂Ω

ψu dx = 0 ⇒ u = 0 on ∂Ω .

Conclusion : u = uΩ.



Cea’s method : the Dirichlet case (V)

Step 2 : Identification of p :

∀v ∈ H1(Rd),

∫Ω

j ′(u) · v dx +

∫Ω

∇v · ∇p dx +

∫∂Ω

(λv − ∂v

∂np

)ds = 0.


∀ψ ∈ C∞c (Ω),

∫Ω

∇p · ∇ψ dx +

∫Ω

j ′(u) · ψ dx = 0

⇒ −∆p = −j ′(uΩ) in Ω .

Now taking v as a C∞ function ψ and using Green’s formula :

∀ψ ∈ C∞c (Rd),

∫∂Ω

∂p

∂nψ ds +

∫∂Ω

(λψ − ∂ψ

∂np

)ds = 0.



Cea’s method : the Dirichlet case (VI)

Step 2 (continued) :

Varying the normal trace ∂ψ∂n

while imposing ψ = 0 on ∂Ω, one gets :

p = 0 on ∂Ω .

Conclusion : p = pΩ, solution to

−∆p = −j ′(uΩ) in Ω

p = 0 on ∂Ω

In addition, varying the trace of ψ on ∂Ω while imposing ∂ψ∂n

= 0 :

λΩ = − ∂pΩ∂n

on ∂Ω.



Cea’s method : the Dirichlet case (VII)

Step 3 : Calculation of the shape derivative J ′(Ω)(θ) :

We go back to the fact that :

∀q, µ ∈ H1(Rd), L(Ω, uΩ, q, µ) =

∫Ω

j(uΩ) dx .

Differentiating with respect to Ω yields, for all θ ∈W 1,∞(Rd ,Rd) :

J ′(Ω)(θ) =∂L∂Ω

(Ω, uΩ, q, µ)(θ) +∂L∂v

(Ω, uΩ, q, µ)(u′Ω(θ)),

where u′Ω(θ) is the Eulerian derivative of Ω 7→ uΩ.

Taking q = pΩ, µ = λΩ produces, since ∂L∂v

(Ω, uΩ, pΩ, λΩ) = 0 :

J ′(Ω)(θ) =∂L∂Ω

(Ω, uΩ, pΩ, λΩ)(θ).



Cea’s method : the Dirichlet case (VIII)

Again, this (partial) derivative amounts to the shape derivative of a functional of theform :

Ω 7→∫

Ω

f (x) dx ,

where f is a fixed function.

Using the derivative of Ω 7→∫

Ωf (and after some calculation), we end up with :

∀θ ∈W 1,∞(Rd ,Rd), J ′(Ω)(θ) =

∫∂Ω

(j(uΩ)− ∂uΩ

∂n

∂pΩ

∂n

)θ · n ds,


Numerics and applications to Fluid Mechanics Reminders in Fluid Mechanics

Contents






Reminders on the PDE in Fluid Mechanics

Definition

Let Ω be a bounded open set in Rd with a smooth boundary and let u and v , two vectorfields in H1(Ω,Rd).

We define the stretching tensor ε of the vector field u by

ε(u) =1

2(∇u + (∇u)T ) =

(1

2

(∂ui∂xj

+∂uj∂xi

))1≤i,j≤d

.

We define the doubly contracted product of two stretching tensors ε(u) and ε(v) by

ε(u) : ε(v) =1

4

d∑i,j=1

(∂ui∂xj

+∂uj∂xi

)(∂vi∂xj

+∂vj∂xi

).

|ε(u)|2 = ε(u) : ε(u).

if p ∈ L2(Ω) stands for the fluid pressure at every point of Ω, if u denotes the fluidvelocity at every point of Ω, then one defines the stress tensorσ(u, p) ∈ L2(Ω, Sd(R)) by

σ(u, p) = −pId + 2µε(u).




Properties of these tensors

Lemma

Let Ω, an open set of Rd . For every u ∈ H1(Ω,Rd), one has

‖ε(u)‖L2(Ω) ≤ ‖∇u‖L2(Ω) et ‖ div(u)‖L2(Ω) ≤ d‖∇u‖L2(Ω).

Proof : using the well-known inequality (a + b)2 ≤ 2(a2 + b2) valid for all real numbers a,b and by definition ε(u), there holds

‖ε(u)‖2L2(Ω) =

1

4

∫Ω

∑1≤i,j≤d

(∂ui∂xj

+∂uj∂xi

)2

dx ≤ 1

2

∑1≤i,j≤d

∫Ω

((∂ui∂xj

)2

+

(∂uj∂xi

)2)

dx

=1

2(‖∇u‖2

L2(Ω) + ‖∇u‖2L2(Ω)),

whence the result.




Properties of these tensors

Lemma

Let Ω, an open set of Rd . For every u ∈ H1(Ω,Rd), one has

‖ε(u)‖L2(Ω) ≤ ‖∇u‖L2(Ω) et ‖ div(u)‖L2(Ω) ≤ d‖∇u‖L2(Ω).

Proof (continued) :Similarly, one has

‖ div(u)‖2L2(Ω) =

∫Ω

∑1≤i≤d

∂ui∂xi

2

dx =∑

1≤i,j≤d

∫Ω

∂ui∂xi

∂uj∂xj

dx

≤ 1

2

∑1≤i,j≤d

∫Ω

((∂ui∂xi

)2

+

(∂uj∂xj

)2)

dx =2d

2

d∑i=1

∫Ω

(∂ui∂xi

)2

dx

≤ d∑

1≤i,j≤d

∫Ω

(∂ui∂xj

)2

dx = d‖∇u‖2L2(Ω).




Theorem (Korn inequality)

Let Ω be an open set in Rd of class piecewise C1. Then, there exists a constant CΩ > 0such that for every function u ∈ H1(Ω,Rd), one has

‖u‖H1 ≤ CΩ(‖u‖L2(Ω) + ‖ε(u)‖L2(Ω)).

Introduce the functional spaces

L20(Ω) = p ∈ L2(Ω) |

∫Ω

p dx = 0

V(Ω) = ϕ ∈ D(Ω,Rd) | div(ϕ) = 0

V (Ω) = v ∈ H10 (Ω,Rd) | div(v) = 0 = V(Ω)

H10 (Ω,Rd )

Theorem (De Rham lemma)

Let Ω be a Lipschitz bounded connected open set in Rd . Let f ∈ H−1(Ω,Rd) s.t.

〈f , ϕ〉H−1,H10

= 0 ∀ϕ ∈ V(Ω).

Then, there exists a unique function p in L20(Ω) such that f = ∇p.




The Navier-Stokes system−µ4u +∇p + (u · ∇)u = f x ∈ Ω,div u = 0 x ∈ Ωu = u0 x ∈ Eu = 0 x ∈ Γ−pn + 2µε(u) · n = −p0n x ∈ S ,

where p0 ∈ R and u0 ∈ H1/2(E ,Rd).

Theorem (Existence and uniqueness)

Assume that u0 ∈ H10 (E ,Rd) and f ∈ L3/2(Ω,Rd).

There exist µ0 > 0 and ε0 > 0 such that if µ ≥ µ0 or if ‖u0‖H10 (E) ≤ ε0, then the Navier-

Stokes problem above has a unique solution (u, p) ∈ H1(Ω)×L2(Ω). Moreover, there existsC > 0 depending only on Ω s.t. the solution u verifies

‖u‖2H1

0 (Ω) ≤ C(‖u0‖2H1

0 (E) + ‖u0‖H10 (E)) + ‖f ‖2

L3/2(Ω).




Variational formulation of this problem

Introduce the functional spaces :

W0(Ω)def=

(v , q) ∈ H1(Ω,Rd)× L2(Ω) : v = 0 on E ∪ Γ

.

Zu0 (Ω)def=

(v , q) ∈ H1(Ω,Rd)× L2(Ω) : v = u0 on E et v = 0 sur Γ

,

endowed with the usual associated topologies.

Set p = p − p0. The variational formulation of the previous Navier-Stokes system writesFind (u, p) ∈ Zu0 (Ω) s.t. ∀(w , ψ) ∈W0(Ω),∫

Ω

(2µε(u) : ε(w) + (u · ∇)u · w − p div w) dx = 0∫Ω

ψdiv u dx = 0.


Numerics and applications to Fluid Mechanics Shape optimization in Fluid Mechanics

Contents






A toy problem : the optimal shape of a duct

Problem and modeling

Consider a fluid driven by the previousNavier-Stokes system, of viscosity µ flowing insidea cannula-shaped pipe/duct. For instance, we lookfor the optimal shape of a pipeline.

The optimal design problem writes infΩ∈Oad

J(Ω) where J(Ω) = 2µ

∫Ω

|ε(u)|2 dx ,

with

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

Oad is the set of admissible shapes, for instance, E (inlet) and S (outlet) are fixedand we look for the lateral boundary such that Ω open connected subset of Rd with|Ω| = V0 (given).




Computation of the shape derivative (I)

! Dirichlet boundary conditions on the lateral free boundary.

! Replacing p by p − p0 leads to assume that the right-hand side on S is 0.

Let us use Cea’s method to compute the derivative of J. Introduce the Lagrangian function

L(Ω, u, pv , q) = 2µ

∫Ω

|ε(u)|2 dx − P(Ω, u, pv , q),

where

P(Ω, u, pv , q, ν) =

∫Ω

(2µε(u) : ε(v) + [(u · ∇)u]v − p div(v)) dx −∫

Ω

div(u)q dx

+

∫Γ

µ · u ds




Computation of the shape derivative (II)

For a fixed shape Ω, we look for the saddle points of the functional L. The first-ordernecessary conditions are :

∀δp ∈ C∞0 (Ω,Rd),∂L∂p

(Ω, u, pv , q, ν)(δp) =

∫Ω

δp div(v) = 0.

and

∀δu ∈ C∞0 (Ω,Rd),∂L∂u

(Ω, u, pv , q, ν)(δu) = 0.

This last equation rewrites

2µ

∫Ω

ε(u) : ε(δu) dx −∫

Ω

(2µε(δu) : ε(v) + [(δu · ∇)u + (u · ∇)(δu)]v) dx

−∫

Ω

div(δu)q dx +

∫Γ

µ · δu ds = 0.




Computation of the shape derivative (III)

We will use the following useful integration by parts formula.

Theorem (Existence and uniqueness)

Let y ∈ H1(Ω,Rd) and z ∈ H2(Ω,Rd). Then,

2

∫Ω

ε(z) : ε(y) dx = −∫

Ω

(4z +∇ div z) · y dx + 2

∫∂Ω

ε(z)n · y ds.

Step 1 : Identification of u : similar to what was done in the elliptic case. In particular,

the function ∂L∂µ

vanishes at the saddle point yielding to u = 0 on ∂Ω.




Computation of the shape derivative (IV)

Step 2 : Identification of the adjoint state : We are now in position to identify the adjointstate.

∀δp ∈ C∞0 (Ω,Rd),∂L∂p

(Ω, u, pv , q, ν)(δp) =

∫Ω

δp div(v) = 0

yieldsdiv v = 0 in Ω.

Using

∀δu ∈ C∞0 (Ω,Rd),∂L∂u

(Ω, u, pv , q, ν)(δu) = 0.

Exercise : integrating by parts this relation, and making δu first with compact support inΩ and then varying on Γ yields that (v , q), solves the following PDE :

− µ4v + (∇u)>v − (∇v)u +∇q = −2µ4u x ∈ Ωdiv v = 0 x ∈ Ωv = 0 x ∈ E ∪ Γ−qn + 2µε(v) · n + (u · n)v − 4µε(u) · n = 0 x ∈ S .

→ Boundary conditions obtained exactly as in the elliptic case.




Computation of the shape derivative (V)

Assume for the moment that the adjoint system is well posed on the space

W0(Ω) =

(v , q) ∈ (H1(Ω))3 × L2(Ω) : v = 0 sur E ∪ Γ.

Recall that

J(Ω) = 2µ

∫Ω

|ε(u)|2 dx

According to the chain’s rule shape derivative formula, we claim that

J ′(Ω)(θ) = 4µ

∫Ω

ε(u) : ε(u′) dx + 2µ

∫Γ

|ε(u)|2(θ · n) ds.

Proposition

Assume that Ω is of class C2. One has

J ′(Ω)(θ) = 2µ

∫Γ

(ε(u) : ε(v)− |ε(u)|2

)(θ · n) ds.




Proof : according to the “chain derivation rules”, there holds

J ′(Ω)(θ) = 4µ

∫Ω

ε(u) : ε(u′) dx + 2µ

∫Γ

|ε(u)|2(θ.n) ds

where u′ denotes the unique solution of the system

−µ4u′ +∇u · u′ +∇u′ · u +∇p′ = 0 x ∈ Ωdiv u′ = 0 x ∈ Ωu′ = 0 x ∈ E

u′ = −∂u∂n

(θ · n) x ∈ Γ

−p′n + 2µε(u′) · n = 0 x ∈ S .

Using the Green formula, we infer

J ′(Ω)(θ) = 4µ

∫Ω

ε(u) : ε(u′) dx + 2µ

∫Γ

|ε(u)|2(θ.n) ds

= −2µ

∫Ω

((4u +∇ div u) · u′) dx + 4µ

∫∂Ω

ε(u)n · u′ ds

+2µ

∫∂Ω

|ε(u)|2(θ · n) ds




Let us multiply the main equation of the adjoint state by u′ and then integrate on Ω. Onegets

−µ∫

Ω

4v · u′ dx +

∫Ω

∇q · u′ dx +

∫Ω

(∇u)>v · u′ dx

−∫

Ω

(∇v)u · u′ dx = −2µ

∫Ω

4u · u′ dx .

Integrating by parts and using the boundary conditions on u′ and v yields∫Ω

(2µε(u′) : ε(v)− (∇v)u′ · u + (∇u′)u · v

)dx −

∫S

σ(v , q)n · u′ ds

+

∫S

((u · v)(u′ · n)− (u · n)(u′ · v)

)ds −

∫Γ

σ(v , q)n · u′ ds = −2µ

∫Ω

4u · u′ dx .




In a similar way, let us multiply the main equation of the PDE satisfied by u′ by v andthen integrate on Ω. One gets

−µ∫

Ω

4u′ · v dx +

∫Ω

∇p · v dx +

∫Ω

∇u′ · u · v dx +

∫Ω

∇u · u′ · v dx = 0.

Integrating by parts and using the boundary conditions on u′ and v yields∫Ω

(2µε(u′) : ε(v) + (∇u′)u · v − (∇v)u′ · u

)dx

+

∫S

(−σ(u′, p)n · v + (u · v)(u′ · n)

)ds = 0.




Let us come back to the computation of the shape derivative :

J ′(Ω)(θ) = −2µ

∫Ω

((4u +∇ div u) · u′) dx + 4µ

∫∂Ω

ε(u)n · u′ ds

+2µ

∫∂Ω

|ε(u)|2(θ · n) ds

= A + 4µ

∫∂Ω

ε(u)n · u′ ds + 2µ

∫∂Ω

|ε(u)|2(θ · n) ds,

where we have set A := −2µ

∫Ω

((4u +∇ div u) · u′) dx .

According to the previous computations, we claim that

A =

∫Γ∪S

(qn − 2µε(v)n) · u′ ds −∫S

(u · n)(v · u′) ds.




Consequently and since (v , q) solves the adjoint state, one has

J ′(Ω)(θ) =

∫Γ∪S

(qn − 2µε(v)n) · u ds −∫S

(u · n)(v · u) ds

+4µ

∫S∪Γ

ε(u)n · u ds + 2µ

∫Γ

|ε(u)|2(θ · n) ds

=

∫Γ

(qn − 2µε(v)n + 4µε(u)n) · u ds + 2µ

∫Γ

|ε(u)|2(θ · n) ds

= −∫

Γ

((qn − 2µε(v)n + 4µε(u)n) · ∂u

∂n+ 2µ|ε(u)|2

)(θ · n) ds

This is not exactly the expected form. One can use a “cosmetic” lemma to conclude.

Lemma

Let x et y , two regular vector fields of R3 s.t. x |Γ = y |Γ = 0 and div x = div y = 0 in Ω. Let

θ ∈W 1,∞(R3,R3). Thus, there holds on Γ :

1 (∇x)θ · n = ((∇x)n · n)(θ · n) = 0,

2 ε(x) : ε(y) = ε(x) : (ε(y)(n ⊗ n)) = (ε(x)n) · (ε(y)n),

3 (ε(x)n) · ((∇y)θ) = (ε(x)n) · ((∇y)n)(θ · n) = (ε(x)n) · (ε(y)n)(θ · n).

with the convention that n ⊗ n =∑3

i,j=1 ninj .Y. Privat (CNRS & Univ. Paris 6) Master 2 - UPMC (2015) feb. 2015 67 / 81



The first equality proves that qn · ∂u∂n

= 0 sur Γ.

The first identity shows that Γ, ε(u)n · ∂u∂n

= |ε(u)|2.

Finally, using the second and the third identity allows to show that on Γ,

(ε(v)n) · ∂u∂n

= ε(u) : ε(v).

As a consequence, the shape gradient of J writes :

∇J(Ω) = 2µ(ε(u) : ε(v)− |ε(u)|2

)n.



On the well-posed character of the adjoint problem

Recall that

W0(Ω) =

(v , q) ∈ (H1(Ω))3 × L2(Ω) : v = 0 sur E ∪ Γ.

Proposition

The variational formulation of the adjoint state writesFind (v , q) ∈W0(Ω) s.t. : ∀(w , ψ) ∈W0(Ω),∫

Ω

(2µε(v) : ε(w) + (∇w)u · v + (∇u)w · v − q div w) dx = 4µ

∫Ω

ε(u) : ε(w) dx∫Ω

∇ψ · v dx = 0.




Proof : somme hints.Consider w ∈ C∞0 (Ω) s.t.∫

Ω

(−µ4v + (∇u)>v − (∇v)u +∇q + 2µ4u

)· w dx

+

∫S

(−qn + µε(v)n + (u · n)v − 2µε(u)n) · w ds = 0.

Applying Green formula yields∫S

(w · v)(u · n) ds −∫

Ω

(∇v)u · w dx =

∫Ω

(∇w)u · v dx ,

and similarly, ∫S

q(w · n) ds −∫

Ω

∇q · w dx =

∫Ω

q div w dx .

Finally, a change of index shows that

∫Ω

(∇u)>v · w dx =

∫Ω

(∇uw) · v dx .




Thus, the integration by parts formula allows to write

−µ∫

Ω

(4v +∇ div v) · w dx + 2µ

∫∂Ω

ε(v)n · w ds = 2µ

∫Ω

ε(v) : ε(w) dx

and − µ∫

Ω

4u · w dx + 2µ

∫S

ε(u)n · w ds = 2µ

∫Ω

ε(u) : ε(w) dx .

Using a standard density argument yields that for (w , ψ) ∈W0(Ω), there holds∫

Ω

(2µε(v) : ε(w) + (∇w)u · v + (∇u)w · v − q div w) dx = 4µ

∫Ω

ε(u) : ε(w) dx∫Ω

∇ψ · v dx = 0,

which concludes the proof.




Theorem (Existence and uniqueness of the adjoint state)

Assume that d = 2 ou d = 3 and that Ω is of class C2. Introduce (v , q), solution of theadjoint state

− µ4v + (∇u)>v −∇vu +∇q = −2µ4u x ∈ Ωdiv v = 0 x ∈ Ωv = 0 x ∈ E ∪ Γ−qn + 2µε(v) · n + (u · n)v − 4µε(u) · n = 0 x ∈ S .

(2)

If the viscosity µ is large enough, then the adjoint problem has a unique solution (v , q) ∈H1(Ω)× L2(Ω).




Proof : without loss of generality, assume that d = 3. To prove the existence and uniquenessof a solution, let us apply Lax-Milgram theorem.Introduce the functional spaces :

D(Ω) : space of C∞ real valued functions having a compact support in Ω.

V(Ω) = u ∈ D(Ω) | div u = 0V (Ω) = completion of V in the space u ∈ H1(Ω) | u = 0 on E ∪ Γ.

Remark : note that one has also V (Ω) = u ∈ H1(Ω) : div u = 0 and u = 0 on E ∪ Γ.According to De Rham lemma, let us work on the space V(Ω). Using the variationalformulation of the adjoint state, v satisfies in a distributional sense

∀w ∈ V(Ω), α(v ,w) = 〈`,w〉,

where α and ` are respectively the bilinear and linear forms :

α(v ,w) =

∫Ω

(2µε(v) : ε(w) + (∇w)u · v + (∇u)w · v) dx

〈`,w〉 = 4µ

∫Ω

ε(u) : ε(w) dx ,

whenever it exists.Y. Privat (CNRS & Univ. Paris 6) Master 2 - UPMC (2015) feb. 2015 68 / 81



Proof (continued) : To apply Lax-Milgram theorem, it remains to show that α is continuous,coercive on V (Ω)2, and that ` is continuous on V (Ω).

α is continuous in H1. The first integral term is dominated by the product of theH1-norms of v and w using the equivalence of the norms ‖∇‖L2 and ‖ε(·)‖L2 in H1

0

(Korn inequality) and the Cauchy-Schwarz inequality. Let us show how to estimatethe second and the third integral terms. There holds∣∣∣∣∫

Ω

(∇w)u · v dx

∣∣∣∣ ≤ ∑1≤i,j≤3

∫Ω

∣∣∣∣∂wi

∂xjujvi

∣∣∣∣ dx ≤ 1

2‖∇w‖2

L2 +1

2‖u‖2

L4‖v‖2L4 .

The conclusion follows by using that the injection H1(Ω) → L4(Ω) is continuoussince d ≤ 3.

` is continuous. It is enough to write that for all v ∈ H1(Ω,R3) and according to theCauchy-Schwarz inequality, there holds

|〈`, v〉| ≤ 4µ‖ε(v)‖L2(Ω,R3)‖ε(u)‖L2(Ω,R3).

We conclude by using the Korn inequality.




Proof (continued) :

α is coercive in H1. Similar computations prove that for all v ∈ H1(Ω,R3) :

α(v , v) ≥ 2µ

∫Ω

|ε(v)|2 dx − ‖u‖2H1(Ω,R3)‖v‖

2H1(Ω,R3).

Now, using one more time the Korn inequality, there exists C > 0 (depending onlyon Ω) s.t.

α(v , v) ≥(

2Cµ− ‖u‖2H1(Ω,R3)

)‖v‖2

H1(Ω,R3).

α is then coercive provided that µ be chosen large enough.

Lax-Milgram theorem shows the existence of v ∈ H1(Ω,R3) satisfying the variationalformulation in V(Ω) and then by density in V (Ω).The use of De Rham lemma yields the existence of a distribution q ∈ L2(Ω) solution ofthe adjoint state.The uniqueness of q is obvious, according to the boundary conditions on the outlet S .indeed, it there would exist two solutions q1 and q2, and since the velocity v is unique,then the function q1 − q2 would verify at the same time ∇(q1 − q2) = 0 in Ω and p1 = p2

on S .


Numerics and applications to Fluid Mechanics Numerical aspects

Contents






The generic numerical algorithm

Gradient algorithm : Start from an initial shape Ω0,For n = 0, ... convergence,

1 Compute the state uΩn (and possibly the adjoint pΩn ) of the consideredPDE system on Ωn.

2 Compute the shape gradient J ′(Ωn) thanks to the previous formula, andinfer a descent direction θn for the cost functional.

3 Advect the shape Ωn according to this displacement field for a smallpseudo-time step τ n, so as to get

Ωn+1 = (I + τ nθn)(Ωn).



One possible implementation

Each shape Ωn is represented by a simplicial mesh T n (i.e. composed of triangles in2d and of tetrahedra in 3d).

The Finite Element method is used on the mesh T n for computing uΩn (and pΩn )The descent direction θn is then calculated using the theoretical formula for theshape derivative of J(Ω).

The shape advection step Ωn (I+τnθn)7−→ Ωn+1 is performed by pushing the nodes of T n

along τ nθn, to obtain the new mesh T n+1.

Deformation of a mesh by relocating its nodes to a prescribed final position.



Numerical examples (I)

In the context of linear elasticity, one aims at minimizing the compliance C(Ω) of acantilever beam :

C(Ω) =

∫Ω

Ae(uΩ) : e(uΩ) dx .

An equality constraint on the volume Vol(Ω) of shapes is imposed.

D

N

g

Minimization of the compliance of a cantilever (from [?] ; code available on [?]).



Numerical examples (II)

In the context of fluid mechanics (Stokes equations), one aims at minimizing theviscous dissipation D(Ω) in a pipe :

D(Ω) = 2µ

∫Ω

|ε(uΩ)|2 dx .

A volume constraint is imposed by a fixed penalization of the function D(Ω) - i.e.the minimized function is D(Ω) + `Vol(Ω), where ` is a fixed Lagrange multiplier.

in

out

Minimization of the viscous dissipation inside a pipe.



Numerical examples (II)

Still in fluid mechanics, minimization of the viscous dissipation D(Ω) in a doublepipe.

A volume constraint is imposed by a fixed penalization of the function D(Ω).

in

uin out

Minimization of the viscous dissipation inside a double pipe.



Numerical issues and difficulties (I)

I - The difficulty of mesh deformation :

When the shape is explicitly meshed, an update of the mesh is necessary at eachstep Ωn 7→ (I + θn)(Ωn) = Ωn+1 : the new mesh T n+1 is obtained by relocating eachnode x ∈ T n to x + τ nθn(x).

This may prove difficult, partly because it may cause inversion of elements, resultingin an invalid mesh.

Pushing nodes according to the velocity field may result in an invalid configuration.

For this reason, mesh deformation methods are generally preferred for accounting for‘small displacements’.



Numerical issues and difficulties (II)

II - Velocity extension :

A descent direction θ = −vΩn from a shape Ω is inferred from the shape derivativeof J(Ω) :

J ′(Ω)(θ) =

∫∂Ω

vΩ(θ · n) ds.

The new shape (I + θ)(Ω) only depends on these values of θ on ∂Ω.

For many reasons, in numerical practice, it is crucial to extend θ to Ω (or even Rd)in a ‘clever’ way.(for instance, deforming a mesh of Ω using a ‘nice’ vector field θ defined on thewhole Ω may considerably ease the process)

The ‘natural’ extension of the formula θ = −vΩn, which is only legitimate on ∂Ωmay not be a ‘good’ choice.



Numerical issues and difficulties (III)

III - Velocity regularization :

Taking θ = −vΩn on ∂Ω may produce a very irregular descent direction, because ofnumerical artifacts arising during the finite element analyses.an inherent lack of regularity of J′(Ω) for the problem at stake.

In numerical practice, it is often necessary to smooth this descent direction so thatthe considered shapes stay regular.

Irregularity of the shape derivative in the very sensitive problem of drag minimization of an airfoil (Taken from [?]). In one iteration, using the unsmoothedshape derivative of J(Ω) produces large undesirable artifacts.



Numerical issues and difficulties (IV)

A popular idea : extend AND regularize the velocity field

Suppose we aim at extending the scalar field vΩ : ∂Ω→ R to Ω.

Idea : (≈ Laplacian smoothing) Trade the ‘natural’ inner product over L2(∂Ω) for amore regular inner product over functions on Ω.

Example : Search the extended / regularized scalar field V as :

Find V ∈ H1(Ω) s.t. ∀w ∈ H1(Ω),

α

∫Ω

∇V · ∇w dx +

∫Ω

Vw dx =

∫∂Ω

vΩw ds.

The regularizing parameter α controls the balance between the fidelity of V to vΩ

and the intensity of smoothing.



Numerical issues and difficulties (IV)

The resulting scalar field V is inherently defined on Ω and more regular than vΩ.

Multiple other regularizing problems are possible, associated to different innerproducts or different function spaces.

A similar process allows :to extend vΩ to a large computational box D (an inner product over functions definedon D is used),to extend the vector velocity θ = −vΩn to Ω / D (an inner product over vectorfunctions is used, e.g. that of linear elasticity).



Numerical issues : moral conclusion

The choice of a numerical method for shape optimization has to reach a tradeoffbetween numerical accuracy and robustness :

The more accurate the representation of the boundaries of shapes, the moreaccurate the mechanical analyses performed on shapes (computation of uΩn ,pΩn , etc...), and the more accurate the computation of descent directions.

... But the more tedious and error-prone the advection step between shapesΩn 7→ Ωn+1.



Numerical results : optimal shape of a cannula


theoretical and numerical aspects for incompressible ...frey/cours/upmc/nm... · sethian j.a.,level...

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