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

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

PascalFreyandYannickPrivat

(Slides by CharlesDapogny, PascalFreyand YannickPrivat)

CNRS & Univ. Paris 6

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Session 1: tools for shape optimization in Fluid Mechanics (optimization) and existence of optimal shapes

Session 2 and 3 : optimality condition (shape derivative) Session 4 and 5 : existence of optimal shapes, application in Fluid Mechanics and algorithms

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

Joining me: [email protected]

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

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Outlines of the lesson

1 Reminders in (in)finite-dimensional optimization problems

2 Several examples of shape optimization problems

3 Generalities

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3 Generalities

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Reminders in (in)finite-dimensional optimization problems

Finite dimensional optimization problems: basics

Letf :K⊂Rⁿ−→Rbe continuous andK⊂Rⁿ. Consider the optimization problem

x∈Kinf f(x)

Existence issues

Theorem (Existence)

Assume the existence ofx0 ∈Rⁿ such that the set{f ≤f(x0)}be bounded. Then, the problem above has at least a solutionx^∗.

! The assumption above is satisfied wheneverK is compactorK closed andf coercive, i.e.

f(x)−−−−−−→

kxk→+∞ +∞

! Example:

inf

(x,y)∈Kf(x,y) =x⁴+y⁴−x² with K ={(x,y)∈R²,x+y ≤4}.

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What about infinite dimensional spaces?

Consider the Hilbert space`²(R) =

x= (xn)n∈N∈R^N|P+∞

n=0x_n²<+∞ ,endowed with the inner producthx,yi=P+∞

n=0xnyn.

The following optimization problem has no solution:

inf

x∈`²(R)

f(x) with f(x) =

kxk²−12

+

+∞

X

n=0

xn²

n+ 1

The space`²(R) is closed, the functionf is continuous, coercive, BUT the problem above has no solution.

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Finite dimensional optimization problems: basics

Uniqueness issues Convex sets and functions

1 A setK⊂Rⁿis said convex iff for all (x1,x2)∈K²andt∈[0,1], one has tx1+ (1−t)x2∈K.

2 LetK⊂Rⁿ be convex. The functionf :K −→Ris saidconvexiff

∀(x1,x2)∈K², ∀t∈[0,1], f(tx1+ (1−t)x2)≤tf(x1) + (1−t)f(x2).

f isstrictly convexif the ineq. above is strict forx 6=y,t∈]0,1[.

strictly convex function

non convex function

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Uniqueness issues

Theorem.

Consider the problem

x∈Kinf f(x)

withf andK convex (also true in infinite dim). Then,

1 every local minimum is global.

2 iff is strictly convex, there exists at most one minimizer.

Important example: the ”quadratic” function f : Rⁿ −→ R

x 7−→ f(x) = ¹₂hAx,xi − hb,xi+c, withA∈ Sn(R),b∈Rⁿandc∈R.

The functionf is (strictly) convex iff Hessf(x) =Ais positive (semi) definite.

More precisely, ifA∈ S_n⁺⁺(R), one has

∃λ >0 | hAx,xi ≥λkxk².

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Finite dimensional optimization problems: basics

Optimality conditions

Letf :Rⁿ →Rbe differentiable and the functionsh:Rⁿ →R^p andg :Rⁿ −→R^q of classC¹.

Consider the problem inf

x∈Kf(x) where K ={x∈Rⁿ, h(x) = 0 andg(x)≤0}.

Theorem (Karush-Kuhn-Tucker)

Letx^∗ be a local minimum of the problem above. We assume that the constraints are qualified atx. Then, there exists (λ1,· · ·, λp)∈R^pet (µ1,· · ·, µq)∈R^q+such that

∇f(x^∗) +

p

X

i=1

λi∇hi(x^∗) +

q

X

j=1

µj∇gj(x^∗) = 0, h(x^∗) = 0 andg(x^∗)≤0,

µjgj(x^∗) = 0,∀j∈ {1,· · ·,q}(complementary slackness).

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Constraints qualification

Proposition

Letx^∗∈K and introduce the active set

I(x^∗) ={1≤j≤q|gj(x^∗) = 0}.

If

either gj withj∈I(x^∗) andhare affine, or the family{∇gj(x^∗),∇hi(x^∗)}1≤i≤p

j∈I(x^∗)

is made of positive linearly independent vectors (Mangasarian-Fromovitz),

or for each subset of{∇gj,∇hi}1≤i≤p j∈I(x^∗)

, the rank at the vicinity ofx^∗is constant, then, the constraints are qualified atx^∗.

Two tell-tale examples:

infx

x∈Retx²= 0, and

infhAx,xi

kxk2≤1, withA∈Sn⁺(R).

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Several examples of shape optimization problems

Sommaire

3 Generalities

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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 ?

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Isoperimetric problems

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).

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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)

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Aerodynamic optimization of airplane

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

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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.

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Optimal shape of a duct

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

|Ω|=V0(given).

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3 Generalities

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Generalities

Mathematical framework

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).

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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-section S is parametrized by its thickness function h : Ω→R.

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Generalities

Various settings for shape optimization (II)

! 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.

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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.

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Generalities

Various settings for shape optimization (IV)

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 ascharacteristic functionsχΩ:D→ {0,1}.

⌦

Optimizing a shape by acting on its topology.

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! 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.

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Generalities

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.

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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. . .