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Theoretical and Numerical Issues of Incompressible Fluid Flows

Chapter 7: Shape optimization

Instructor: Pascal Frey

Sorbonne Université, CNRS4, place Jussieu, Paris

University of Tehran, 2018

Section 7.1Framework

Introduction

• the main target of shape optimization is to provide a common and systematic frameworkfor optimizing structures described by various possible physical or mechanical models

• this topic has become increasingly popular in academics and industry, partly becauseraw materials are becoming increasingly expensive, hence making it necessary to opti-mize mechanical parts from the early stages of design

• automatic techniques have started to replace the traditional trial-and-error methodsused by engineers, but still leave room for many forthcoming developments.

• an ancient concern : isoperimetric problems.

Theoretical and numerical issues for Incompressible Fluid Flows University of Tehran, April-May, 2018 3/ 74

Isoperimetric problems

• the problem of DidonThe Aeneid reports that, around 814 BC, Didon landed on the shores of north Africa, andasked a local Berber king, Larbas, a small portion of land to erect a city. King Larbasreplied that he would grant her as much land as she could encompass with an oxhide(the skin of an ox). With a remarkable mathematical intuition, Didon slit the oxhide intovery thin strips, which she glued together, then used to delimit a region of maximumarea, comprising a portion of the seashore, the length of the strip being fixed. This landbecame Carthage and Dido became the Queen.The question was therefore to find the greatest area possible bounded by a line (theshore) with given fixed boundary length.Her strip of oxhide reached the shape of an arc of circle meeting the sea, which was tobecome the boundary of the territory of Byrsa.



• mathematically, this problem is very close to the standard isoperimetric one :Find Ω ∈ R2, solution of the problem

maximize 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. In otherwords, one has to solve for b > a ≥ 0,

supy∈E

∫ bay(x) dx

where

E = y ∈ Y ,∫ ba

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

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



The proof

• Zénodore (2nd century B.C.) proves theisoperimetric inequality in the particularcase where Ω is a polygon.

• Until the 20th century : this result is con-jectured but not proved.

• Steiner (Swiss mathematician of the 19thcentury) publishes a proof, but . . . thisproof is erroneous !

Several examples of shape optimization problems


The proof

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

Y. Privat (CNRS & Univ. Paris 6) Master 2 - UPMC (2015) feb. 2015 4 / 81

• Weierstrass (German mathematician of the 19th century) concludes the proof, by usingmodern tools of calculus of variations;

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


Other optimization problems

• in 1685, Sir Isaac Newton got interested in finding the shape of a body opposing theslightest possible resistance to the motion when immersed in a fluid. Making severaldrastic reductions in the problem (e.g. supposing the shapes of interest are axisymmet-ric), he obtained the shapes:


Other optimization problems

• in 1773, Lagrange formulated the problem of finding the shape of an axisymmetric col-umn of prescribed volume, which guarantees maximal resistance against buckling whensubmitted to axial compression efforts.Using mathematical tools from the calculus of variations, he ended up with the conclu-sion that the cylinder was the optimal shape he was searching for.Unfortunately, he committed several mistakes in his computations, and further devel-opments - notably those of T. Clausen - evidenced that better shapes can be achieved.

• Since those historical examples, shape optimization has been enjoying numerous de-velopments, both in terms of theoretical and numerical techniques. Yet, a huge amountof issues stay unsolved


Framework

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

minΩ∈Uad

J(Ω)

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

• in most mechanical or physical applications, the relevant objective functions J(Ω) de-pend onΩ via a state uΩ, which arises as the solution to a PDE posed onΩ (e.g. a linearelasticity system, or the Stokes equations).


Main methods

The problem of how to perform perturbations of the considered shapes, and thus of how tocompute the sensitivity of the objective function J with respect to the shape, is closelyrelated to this problem of shapes’ description.The following design-sensitivity analysis methods have been extensively considered in theliterature:

• perform a sensitivity analysis of the objective criterion J with respect to perturbationsof the boundary of shapes. The Hadamard’s and speed methods belong to this category.

• perform topological sensitivity analyses, according to which the sensitivity of J with re-spect to the nucleation of infinitesimally small holes inside shapes is evaluated. Similartechniques are also widely used in the fields of imaging or inverse problems.

• when shapes are represented as density functions over a computational domain D, Jdepends on the shape through the values of an associated density function at the nodesof a mesh of D, and a ‘classical’ parameter sensitivity analysis can be performed.


Numerical difficulties

A major difficulty exists on the theoretical side. This problem is that of the non existence ofan optimal shape, which is due to a homogenization phenomenon.

• let D ⊂ Rd a bounded Lipschitz domain. Our goal is to optimize the distribution oftwo materials within D, one of them being thermally conductive, and the other beingthermally insensitive, in such a way that the temperature inD is as close as possible toa constant c when D is heated.

• rigorously speaking, one searches for the shape Ω ⊂ D (standing for the phase filledwith the conductive material) which minimizes the following functional:

J(Ω) =∫

D|uΩ − c|2 dx ,

where c is a constant and uΩ is the unique solution in H10(Ω) to the system:

−∆u = 1 in Ω

u = 0 on ∂Ω



The non existence result is the following.

Theorem 1. For c > 0 small enough, no Lipschitz domain Ω ⊂ D can be a global minimumpoint of J over Uad.

sketch of proof:

1. it is easily proven that, provided c is small enough the whole domainD is not a global minimum point ofthe problem, for J(∅) = c2 is then smaller than J(D).

2. assume that a global minimum point Ω of J exists which is Lipschitz; as Ω ⊆ D there exists a pointx0 ∈ D\Ω lying outside Ω, and let ε > 0 such that d(x0, Ω > ε). Finally, consider the new phaseΩ = Ω ∪ B(x0, ε) for the conductive material, which is nothing but the previous one, augmented witha small disconnected bubble of material. An explicit expression for uΩ can be computed explicitly:

∀x ∈ Ω, uΩ =

uΩ if x ∈ Ω

ε2 − |x− x0|22d

if x ∈ B(x0, ε)

0 if x ∈ D\Ω

3. another computation shows then that J(Ω) < J(Ω) for c > 0 small enough, in contradiction with Ωbeing a minimum point for J .



Understanding the result.

• roughly speaking, the adiabatic heating of a smallamount of conductive material produces a smallpositive temperature, and the larger the region, thelarger the resulting maximal temperature devel-oped in the thermally conductive phase.

• hence, reaching a small temperature c requiresthat the shape of the conductive phase has a largecontact surface with the outer medium, and a smallarea;

• the ’optimal shape’ for Ω would be an infinite collection of infinitesimally small inclu-sions of conductive material (which is not a Lipschitz domain).


Different settings for optimizing the shape

1. parametric optimization

• the considered shapes are described bymeans of a set of physical parameters pii=1,...,N ,typically thickness, curvature radii, etc...

Figure 1: left: description of an airfoil by NURBS; the parameters of the representation are the control points pi.Right: a plate with fixed cross-section S is parametrized by its thickness function h : Ω→ R.



• 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 to accountfor 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.



2. geometric shape optimization

• the topology (i.e. the number of holes in 2d) of theconsidered shapes is fixed

• the boundary ∂Ω of the shapes Ω is the optimiza-tion variable

• geometric optimization allows more freedom thanparametric optimization, since no a priori knowl-edge of the relevant regions of shapes to act on isrequired

Optimization of a shape by performing"free" perturbations of its boundary.



3. topology optimization

• in some applications, the suitable topology ofshapes is unknown, and also subject to optimiza-tion

• it is often preferred not to describe the boundariesof shapes, but to resort to different representationswhich allow for a more natural account of topolog-ical changesfor instance: describing shapes Ω as density func-tions χ : D → [0,1].

Optimizing a shape by acting on itstopology.



• 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, density func-tions, 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, all these different methods for shape optimization share a lot of common fea-tures

• we focus on geometric shape optimization methods.


Section 7.2Shape optimization in mechanics

Shape optimization in structural mechanics

• we consider a structure Ω ⊂ Rd, which is fixed on a part ΓD ⊂ ∂Ω of its boundary, submitted to surface loads g, applied on ΓN ⊂∂Ω, ΓD ∩ ΓN = ∅.

• the displacement vector field uΩ : Ω → Rd isgoverned by the linear elasticity system:

−div(Ae(u)) = 0 in Ω

u = 0 on ΓDAe(u)n = g on ΓNAe(u)n = 0 on Γ := ∂Ω\(ΓD ∪ ΓN)

where e(u) = 12(∇ut +∇u) is the strain tensor

field, and A is the Hooke’s law of the material.

A model problem in linear elasticity

A structure is represented by a bounded open domain Rd, fixed on a part D @ of its boundary, andsubmitted to a load case g (and no body force), to beapplied on N @, D \ N = ;.

The displacement vector field u : ! Rd is gov-erned by the linear elasticity system :8>>><>>>:

div(Ae(u)) = 0 in u = 0 on D

Ae(u).n = g on NAe(u).n = 0 on := @ \ (D [ N)

,

where e(u) = 12(

tru + ru) is the strain tensor field,Ae(u) = 2µe(u) + tr(e(u))I is the stress tensor,and , µ are the Lamé coefficients of the material.

A ‘Cantilever’

The deformed cantilever

PICOF 2012 April 3rd , 2012 4

top: a cantilever beam, bottom: the de-formed cantilever.



• examples of objective functions:

the work of external loads g or compliance C(Ω) of domain Ω:

C(Ω) =∫

ΩAe(uΩ) : e(uΩ)dx =

∫

ΓNg.uΩ ds

a least-square discrepancy between the displacement uΩ and a target displacementu0 (useful when designing micro-mechanisms):

D(Ω) =(∫

Ωω(x)|uΩ − u0|α dx

)1α

where α is a fixed parameter, and ω(x) is a weight factor.



• constraints can be enforced:

on the volume V ol(Ω) =∫

Ω1 dx, or on the perimeter P (Ω) =

∫

∂Ω1 ds of

shapes

on the total stress developped in shapes:

S(Ω) =∫

Ω‖σ(uΩ)‖2 dx

where σ(u) = Ae(u) is the stress tensor

or geometric constraints, e.g. on the minimal and maximal thickness of shapes, onmolding directions, etc... Such constraints play a crucial role when it comes to man-ufacturing shapes.


Shape optimization in fluid mechanics

• an incompressible fluid rests in a domain Ω ⊂ Rd

the flow uin through the input boundary Γin is known a pressure profile pout is imposed on the exit boundary Γout

no slip boundary conditions are considered on the free boundary ∂Ω\(Γin∪Γout)

• the velocity uΩ : Ω → Rd and pressure pΩ : Ω → R of the fluid satisfy Stokesequations:

−div(D(u)) +∇p = f in Ω

div(u) = 0 in Ω

u = uin on Γin

u = 0 on Γ

σ(u)n = −pout on Γout

where D(u) = 12(∇ut +∇u) is the symmetrized gradient of u.



• model problem I: optimization of the shape ofa pipe

the shape subject to optimization is a pipe,connecting the (fixed) input area Γin andoutput area Γout

we are interested in minimizing the totelwork of the viscous forces inside the shape:

J(Ω) = 2µ∫

ΩD(uΩ) : D(uΩ) dx

a constraint on the volume V ol(Ω) of thepipe is enforced.

pipe shape to be optimized.



• model problem II: reconstruction of the shape of an obstacle

an obstacle of unknown shape ω is immersed in a fixed domainD filled by the fluid given a mesure umeas of the velocity uΩ of the fluid inside a small observation areaO, the objective is to reconstruct the shape of ω

the optimized domain is Ω := D\ω, and only the part ∂ω of ∂Ω is optimized. Onethen minimizes the least-square criterion:

J(Ω) =∫

O|uΩ − umeas|2 dx


Difficulty of these problems

• from the modelling viewpoint: difficulty to describe the physical problem at stake by amodel which is relevant (complicated enough), yet tractable (simple enough)

• from the theoretical viewpoint: often, optimal shapes do not exist, and shape optimiza-tion problems enjoy at most local optima

• From both theoretical and numerical viewpoints: the optimization variable is the do-main! Hence the need for of a means to differentiate functions depending on the do-main, and before that, to parametrize shapes and their variations

• On the numerical side: difficulty to represent shapes and their evolutions shape optimization problems may be very sensitive and can be completely domi-nated by discretization errors.


Bibliography - references

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

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

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

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

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

6. Sethian J.A., Level Set Methods and Fast Marching Methods : Evolving Interfaces in Com-putational Geometry, Fluid Mechanics, Computer Vision, and Materials Science, Cam-bridge University Press, (1999).

and many more...


Section 7.3Hadamard’s and Céa’s methods

Hadamard’s method

differentation with respect to the domain

Hadamard’s boundary variation method describesvariations of a reference, Lipschitz domainΩ of theform:

Ω→ Ωθ := (I + θ)(Ω) ,

for ’small’ vector fields θ ∈W1,∞(Rd,Rd).

Lemma 1. for θ ∈ W1,∞(Rd,Rd) such that ‖θ‖W1,∞(Rd,Rd) < 1, the application (I + θ)

is a Lipschitz diffeomorphism.


Hadamard’s method

Definition 1. given a smooth domain Ω, a scalar function Ω → J(Ω) ∈ R is said to beshape differentiable at Ω if the function

W1,∞(Rd,Rd) 3 θ 7→ J(Ωθ)

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

J(Ωθ) = J(Ω) + J ′(Ω)(θ) + o(‖θ‖W1,∞(Rd,Rd))

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

idea: the shape derivative J ′(Ω)(θ) of a regular func-tional J(Ω) depends on the normal component θ · nof the vector field θ.At first order, a tangential vector field θ only results ina convection of the shape Ω and it is expected thatJ ′(Ω)(θ) = 0.


Structure of shape derivatives

Lemma 2. Let Ω be a domain of class C1. Assume that the mapping

C1,∞(Rd,Rd) 3 θ 7→ J(Ωθ) ∈ R

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

Corollary 1. Under the same hypothesis, if θ1, θ2 ∈ C1,∞(Rd,Rd) have the same normalcomponent, θ1 · n = θ2 · n on ∂Ω, then:

J ′(Ω)(θ1) = J ′(Ω)(θ2) .

The shape derivative of many integral objective functionals J(Ω) can be put under theform:

J ′(Ω)(θ) =∫

∂ΩvΩ(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 asmall enough descent step t > 0 yields:

J(Ωtθ) = J(Ω)− t∫

∂Ωv2

Ω ds+ o(t) < J(Ω) .

Theorem 2. Let Ω ⊂ Rd be a bounded Lipschitz domain, and f ∈ W1,1(Rd) be a fixedfunction. Consider the functional:

J(Ω) =∫

Ωf(x) dx

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

∀θ ∈W1,∞(Rd,Rd), J ′(Ω)(θ) =∫

∂Ωf(θ · n) ds .



Figure 2: Fig. 6 Physical intuition: J(Ωθ) is obtained from J(Ω) by adding the blue area, where θ · n > 0, andremoving the red area, where θ · n < 0. The process is ’weighted’ by the integrand function f .

Remark 1. 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 functional V ol(Ω) =

∫

Ω1 dx:

∀θ ∈W1,∞(Rd,Rd), V ol′(Ω)(θ) =∫

∂Ωθ · nds =

∫

Ωdiv(θ) dx .

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


Examples of shape derivatives

Theorem 3. Let Ω0 ⊂ Rd be a bounded, sufficiently regular domain and g ∈W2,1(Rd) bea fixed function. Consider the functional:

J(Ω) =∫

∂Ωg(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 denotes the mean curvature of ∂Ω.


Examples of shape derivatives

• example: the shape derivative of the perimeter P (Ω) =∫∂Ω 1 ds is

P ′(Ω)(θ) =∫

∂Ωκ(θ · n) ds .

• more interesting examples (i.e. of physical interest) are related to PDE constrained shapeoptimization: minimization of functions which depend onΩ via the solution uΩ of a PDE:

J(Ω) =∫

Ωj(uΩ) dx+

∫

∂Ωk(uΩ) ds ,

where, uΩ is the solution of a problem posed on Ω, and j, k are given functions. Thissetting allows to borrow methods from optimal control theory (adjoint techniques, etc.).


More general examples of shape derivative

• consider Laplace equation: the state solution uΩ is solution to the system:

−∆u = f in Ω

u= 0 on ∂Ω (Dirichlet boundary cond.)∂u

∂n= 0 on ∂Ω (Neumann boundary cond.)

where∫

Ωf dx = 0 in the Neumann case.

• the associated variational formulation reads:

∀v ∈ H10(Ω)/H1(Ω),

∫

Ω∇u · ∇v dx−

∫

Ωfv dx = 0

objective: compute the shape derivative of J(Ω) =∫

Ωj(uΩ) dx, where j : R→ R is

a sufficiently smooth function.


Eulerian vs. Lagrangian derivatives

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.There are two ways of looking at this problem:

Eulerian point of view

for a fixed x ∈ Ω, u′Ω(θ)(x) is the deriva-tive of the application

θ 7→ uΩθ(x) .

Introduction Examples Shape derivatives Numerics Other methods

Eulerian and Lagrangian derivatives (I)

The rigorous way to address this problem requires a notion ofdifferentiation of functions 7! u, which to a domain associate afunction defined on . One could think of two ways of doing so:

The Eulerian point of view:For a fixed x 2 , u0

()(x) isthe derivative of the application

7! u(x).

•x

u

u

The Lagrangian point of view:For a fixed x 2 , u()(x) isthe derivative of the application

7! u((I + )(x)).

•x

(x)

• x + (x)

36 / 91

Lagrangian point of view

for a fixed x ∈ Ω, uΩ(θ)(x) is the deriva-tive of the application

θ 7→ uΩθ((I + θ)(x)) .


Eulerian and Lagrangian derivatives (I)

The rigorous way to address this problem requires a notion ofdifferentiation of functions 7! u, which to a domain associate afunction defined on . One could think of two ways of doing so:

The Eulerian point of view:For a fixed x 2 , u0

()(x) isthe derivative of the application

7! u(x).

•x

u

u

The Lagrangian point of view:For a fixed x 2 , u()(x) isthe derivative of the application

7! u((I + )(x)).

•x

(x)

• x + (x)

36 / 91Theoretical and numerical issues for Incompressible Fluid Flows University of Tehran, April-May, 2018 37/ 74


• the Eulerian notion of shape derivative, however more intuitive, is more difficult to de-fine rigorously. In particular, differentiating the boundary conditions satisfied 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 itself tomathematical analysis

• the Eulerian derivative will be defined after the Lagrangian derivative, from the formaluse of chain rule over the expression:

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



Definition 2. the function u : Ω 7→ u(Ω) admits a material or Lagrangian derivative u(Ω)

at a given domain Ω provided the transported function

W1,∞(Rd,Rd) 3 θ 7→ u(θ) := u(Ωθ) (I + θ) ∈ H1(Ω) ,

which is defined in the neighborhood of 0, is diffrentiable at θ = 0.

Definition 3. the function u : Ω 7→ u(Ω) admits a Eulerian derivative u′(Ω)(θ) at a givendomainΩ in the direction θ if it admits a material derivative u(Ω)(θ) atΩ and∇u(Ω)·θ ∈H1(Ω). One defines then: u′(Ω)(θ) = u(Ω)(θ)−∇u(Ω) · θ ∈Wm,p(Ω).

Proposition 1. let Ω ⊂ Rd be a smooth bounded domain and suppose that Ω 7→ u(Ω)

has a Lagrangian derivative u(Ω) at Ω. If j : R → R is regular enough, the functionJ(Ω) =

∫Ω j(u(Ω)) dx is then shape differentiable at Ω and

∀θ ∈W1,∞(Rd,Rd), J ′(Ω)(θ) =∫

Ω(u(Ω)(θ) + θ div(u(Ω))) .

If u(Ω) has a Eulerian derivative u′(Ω) at Ω, one has the ’chain rule’:

J ′(Ω)(θ) =∫

∂Ωj(u(Ω))θ · nds+

∫

Ωj′(u(Ω))u′(Ω)(θ) dx .


Céa’s method

• the philosophy of Céa’s method comes from optimization theory:

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

L(Ω, u, p) =∫

Ωj(u) dx

︸︷︷︸objective function

+∫

Ω(−∆u− f)p dx

︸︷︷︸u=uΩ is enforced as a constraint

by penalization withthe 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.


Céa’s method: the Neumann case

consider the following Lagrangian functional:

L(Ω, v, q) =∫

Ωj(v) dx

︸︷︷︸objective function

where uΩis replaced by v

+∫

Ω∇v · ∇q dx−

∫

Ωfq dx

︸︷︷︸penalization of the constraint v=uΩ∫

Ω(−∆v−f)q dw=0

which is defined for any shape Ω ∈ Uad and for any v, q ∈ H1(R), 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) =∫

Ωj(uΩ) dx = J(Ω) .

for a fixed shape Ω, we search for the saddle points (u, p) ∈ R × R of L(Ω, ·, ·); thefirst-order necessary conditions read:

∀q ∈ H1(Rd),∂L∂q

(Ω, u, p)(q) =∫

Ω∇u · ∇q dx−

∫

Ωfv dx = 0

for all v ∈ H1(Rd),∂L∂v

(Ω, u, p)(v) =∫

Ωj′(u) · v dx+

∫

Ω∇u · ∇p dx = 0



1. identification of u:

∀q ∈ H1(Rd),∫

Ω∇u · ∇q dx−

∫

Ωfq dx = 0

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

∀ψ ∈ C∞c (Ω),∫

Ω∇u · ∇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.



2. identification of p:

∀v ∈ H1(Rd),∫

Ωj′(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∇u · ∇ψ dx+

∫

Ωj′(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

∂nψ ds = 0⇒ ∂p

∂n= 0 on ∂Ω .

conclusion: p = pΩ, solution to

−∆p = −j′(uΩ) in Ω∂p

∂n= 0 on ∂Ω



3. calculation of the shape derivative J ′(Ω)(θ): we go back to the fact that

∀q ∈ H1(Rd), L(Ω, uΩ, q) =∫

Ωj(uΩ) dx

differentiating with respect to Ω yields: for all θ ∈W1,∞(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) .



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

Ω 7→∫

Ωf(x) dx ,

where f is a fixed function

• after some calculation, we end up with: ∀θ ∈W1,∞(Rd,Rd)

J ′(Ω)(θ) =∫

∂Ω(j(uΩ) +∇uΩ · ∇pΩ − fpΩ)θ · nds .

• likewise for Dirichlet conditions, we end up with: ∀θ ∈W1,∞(Rd,Rd)

J ′(Ω)(θ) =∫

∂Ω

(j(uΩ)− ∂uΩ

∂n

∂pΩ

∂n

)θ · nds ,


Numerical aspects

• the generic numerical algorithm reads:

Gradient algorithm: start from an initial shape Ω0,

For n = 0, ... convergence,

1. compute the state uΩn (and possibly the adjoint pΩn) of the considered PDE sys-tem on Ωn

2. compute the shape gradient J ′(Ωn) thanks to the previous formula, and infer adescent direction θn for the cost functional

3. advect the shape Ωn according to this displacement field for a small pseudo-timestep τn, so as to get

Ωn+1 = (I + τnθn)(Ωn) .


Implementing the algorithm

• each shape Ωn is represented by a simplicial mesh Tnh (triangles or tetrahedra)

• the finite elementmethod is used on themeshTnh for computinguΩn (and pΩn). The de-scent direction θn is then calculated using the theoretical formula for the shape deriva-tive of J(Ω)

• the shape advection step Ωn (I+τnθn)7−→ Ωn+1 is performed by pushing the nodes ofTnh along τnθn, to obtain the new mesh Tn+1

h .


One possible implementation

• Each shape n is represented by a simplicial mesh T n (i.e.composed of triangles in 2d and of tetrahedra in 3d).

• The Finite Element method is used on the mesh T n for computingun (and pn) The descent direction n is then calculated using thetheoretical formula for the shape derivative of J().

• The shape advection step n (I+nn)7! n+1 is performed by pushingthe nodes of T n along nn, to obtain the new mesh T n+1.

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

65 / 91

Figure 3: deformation of a mesh by relocating its nodes to a prescribed final position.


Numerical issues

1. difficulty of mesh deformation when the shape is explictly meshed, an update of the mesh is necessary at each step

Ωn 7→ (I+θn)(Ωn) = Ωn+1: the newmesh Tn+1h is obtained by relocating each

node x ∈ Tnh to x+ τnθn(x)

this may prove difficult, partly because it may cause tangling the elements, resultingin an invalid mesh

for this reason, mesh deformation methods are generally preferred for accountingfor ’small displacements’.


Numerical issues


Numerical issues and difficulties (I)

I - The difficulty of mesh deformation:

• When the shape is explicitly meshed, an update of the mesh isnecessary at each step n 7! (I + n)(n) = n+1: the new meshT n+1 is obtained by relocating each node x 2 T n to x + nn(x).

• This may prove difficult, partly because it may cause inversion ofelements, resulting in an invalid mesh.

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

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

69 / 91

Figure 4: pushing nodes according to the velocity field may result in tangled mesh.

2. velocity extension a descent direction θ = −vΩnn from a shapeΩ is inferred from the shape derivativeof J(Ω):

J ′(Ω)(θ) =∫

ΩvΩ(θ · n) ds .

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


for many reasons, it is crucial to extend θ to Ω (or even Rd) in a clever way(e.g. deforming a mesh of Ω using a nice vector field θ defined on the whole Ω mayconsiderably ease the process)

the ’natural’ extension of the formula θ = −vΩnn , which is only legitimate on ∂Ω

may not be a ’good’ choice.

Numerical issues

3. velocity extension and regularization suppose we aim at extending the scalar field vΩ : ∂Ω→ R to Ω

idea: trade the natural inner product over L2(∂Ω) for a more regular inner productover functions on Ω

example: search the extend-regularized scalar field V as: find V ∈ H1(Ω) suchthat ∀v ∈ H1(Ω)

α∫

Ω∇V · ∇w dx+

∫

ΩV w dx =

∫

∂ΩvΩw ds .

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

and the intensity of smoothing.


Practical aspects

• the level set methodthe motion of a domain Ω(t) ⊂ Rd along a ve-locity field v(t, x) ∈ Rd is translated in terms ofa ’level set function’ φ(t, ·) by the level set ad-vection equation:

∂φ

∂t(t, x) + v(t, x) · ∇φ(t, x) = 0

if v(t, x) is normal to the boundary ∂Ω(t):

v(t, x) := V (t, x)∇φ(t, x)

|∇φ(t, x)| ,

the evolution equation rewrites as a Hamilton-Jacobi equation:

∂φ

∂t) + V (t, x)|∇φ(t, x)| = 0 .


Other kinds of representation of shapes: the level set method (II)

The motion of a domain (t) Rd alonga velocity field v(t, x) 2 Rd is translatedin terms of a ‘level set function’ (t, .) bythe level set advection equation:

@

@t(t, x) + v(t, x).r(t, x) = 0

If v(t, x) is normal to the boundary @(t):

v(t, x) := V (t, x)r(t, x)

|r(t, x)| ,

the evolution equation rewrites as aHamilton-Jacobi equation:

@

@t(t, x) + V (t, x)|r(t, x)| = 0

(t) = [(t, .) < 0]

(t + dt) = [(t + dt, .) < 0]

v(t, x)

x•

•

•

77 / 91


Practical aspects

• a fixed computational boxD is meshed once and for all(e.g. with quadrilateral elements)

• each shape Ωn is represented by a level set functionφn, defined at the nodes of the mesh

• as soon as a descent direction θn from Ωn has beencalculated (as a scalar field defined at the nodes of themesh), the advection step Ωn 7→ Ωn+1 = ((I +

τn)θn)(Ωn) is achieved by solving:

∂φ

∂t+ θn|∇φ| = 0 t ∈ (0, Tn), x ∈ D

φ(0, ·) = φn


The level set method in the context of shape optimization (I)

• A fixed computational box D is meshed onceand for all (e.g. with quadrilateral elements).

• Each shape n is represented by a level setfunction n, defined at the nodes of the mesh.

• As soon as a descent direction n from n

has been calculated (as a scalar field definedat the nodes of the mesh), the advection stepn 7! n+1 = (I + nn)(n) is achieved bysolving:

@@t + n|r|= 0 t 2 (0, n), x 2 D(0, ·) = n

74 G. ALLAIRE, F. de GOURNAY, F. JOUVE, A.-M. TOADER

Figure 8. Optimal mast in 2-d: boundary conditions and iterations 6, 11, 16,21 and 100

of a sti material and excluded from optimization. In the formula for J2, thelocalization coecient k(x) is non-zero (equal to 1) only at the boundary and thetarget displacement u0 is (0, 1) on the top boundary, (0,1) on the bottom oneand (0, 0) on the lateral ones. The Lagrange multiplier is ` = 0. Starting from afull domain initialization we perform 500 iterations with the coupling parameterntop = 15 (see Fig. 9). As usual, the convergence is slower than for complianceminimization (see Fig. 10). Furthermore, the computed optimal design is verysensitive to all parameters of the algorithm including the stiness ratio betweenthe weak ersatz material and the true material (which is here equal to 102),the coupling parameter ntop, and the initialization. Dierent choices of theseparameters lead to dierent topologies with similar performances.

Our second example is a gripping mechanism. Fig. 11 shows the boundaryconditions and the target displacement. A small force, parallel to the targetdisplacement in the opposite direction, is also applied on the jaws of the me-

Shape accounted for bya level set description

(from [AlJouToa])

78 / 91Theoretical and numerical issues for Incompressible Fluid Flows University of Tehran, April-May, 2018 52/ 74

Practical aspects

• problem: shapes are not meshed: how to solve a PDE on a shape Ω ?

• solution: approximate the PDE problem posed on Ω by a problem posed on the wholebox D

• example: in the context of linear elasticity, the ersatz material approach (Allaire, Jouve,Toader) consists in filling the void D\Ω with a very soft material with Hooke’s law εA,ε 1

−div(Ae(u)) = 0 ∈ Ω

+ bc≈

−div(AΩe(u)) = 0 in DAΩ = 1ΩA+ (1− 1Ω)εA

+ bc

problem posed on Ω problem posed on D

• we are interested in the optimization of a bridge with respect to its compliance C(Ω)

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



• we consider a bounded domain Ω ⊂ Rd, filled with a linear elastic isotropic materialwith Hooke’s lawA and clamped on a given part ΓD ⊂ ∂Ω, submitted to external loadsg on anotherpart ΓN ⊂ ∂Ω, with ΓD ∩ ΓN = ∅, so that the displacement field u ofthe structure is the unique solution to the system:

−div(Ae(u)) = 0 in Ω

u = 0 &on ΓDAe(u) · n = g on ΓNAe(u) · n = 0 on Γ

where e(u) = (∇u+∇ut)/2 is the linearized strain tensor, γ := ∂Ω\(ΓD ∪ΓN) isthe free boundary and n is the outer unit normal to ∂Ω.

• the problem is to minimize the compliance functional J(Ω) suc that (ΓD∪ΓN) ⊂ ∂Ω

leaving only Γ subject to optimization:

J(Ω) =∫

ΓNg · uΩ ds+ `

∫

Ωdx

where ` is a positive Lagrange multiplier associated to the volume constraint.



• the derivative of J is :

J ′(Ω)(θ) =∫

Γ(`−Ae(uΩ) : e(uΩ)) (θ · n) ds .

• from J ′, a descent direction for J is easily revealed:displacing Ω according to the vector field

θ = τ(Ae(uΩ) : e(uΩ)− `)n

for a small enough fictitious time τ > 0 will yield a new shape with better performancewith respect to J

• we may now put this notion of shape derivative in the general framework of advection,that is, use the algorithm with the velocity field given by:

f(t, x,Ω) = ((Ae(uΩ) : e(uΩ))(x− `))n(x)

to produce an algorithm for optimizing the shape of an initial domain Ω0.



Author's personal copy

374 C. Dapogny et al. / Journal of Computational Physics 262 (2014) 358–378

Fig. 11. (From left to right): Initial, 10th and final (50th) iterations of the optimal mast test-case. Only the boundary ∂Ω of each shape Ω is displayed on theupper range, and only the ‘interior’ part of the associated mesh TΩ of D is displayed on the corresponding cuts of the lower range.

From (10), a descent direction for J is easily revealed: displacing Ω according to the vector field

θ = τ!

Ae(uΩ ) : e(uΩ ) − ℓ"n

for a small enough fictitious time τ > 0 will yield a new shape with better performance with respect to J .We may now put this notion of shape derivative in the general framework of Section 7.1, that is, use the above algorithm

with the velocity field given by:

f (t, x,Ω) =!!

Ae(uΩ ) : e(uΩ )"(x) − ℓ

"n(x) (11)

to produce an algorithm for optimizing the shape of an initial domain Ω0 with respect to J . Note that we have been a bitelusive in writing (11) as such, for the right-hand side only stands for points x ∈ ∂Ω . This expression should actually beextended to the whole space (a tubular neighborhood of ∂Ω is enough in numerical practice); the way such an extensionshould be performed is a topic on its own in shape optimization, and we limit ourselves to referring to [27].

The proposed method is appraised on the so-called optimal mast test case, as depicted in Fig. 11: a structure, embeddedin a T-shaped box D of height 126, and width 40 at the bottom, 80 at the top, made of an isotropic elastic material of Youngmodulus E = 1 and Poisson ratio ν = 0.3, is clamped on its base, and submitted to unit vertical loads g = −ez concentratedon the left and right arms. We minimize the objective function (9) with a Lagrange multiplier ℓ = 20. We run 50 iterationsof the above algorithm; each mesh TΩn (of the whole box) has about 15 000 vertices, and the entire computation takesroughly 40 min on a laptop computer. The decrease in the objective function in the course of the optimization processis displayed in Fig. 12. A noticeable feature of the presented computation is that the topology of the evolving shape hasdramatically changed during the process, which is an inherent (and natural) achievement of the level set description of theshape evolution.

Remark 3. The shape optimization model (9) is admittedly very rough; so to speak, no volume constraint is really imposedsince it only expresses a fixed penalization of the function whose minimization is sought – namely the compliance – withthe volume function

#Ω dx (which is why the volumes of shapes change in the course of the process in the example of

Fig. 11). However, as a slight variation of this model, an update procedure of the Lagrange multiplier ℓ could be though of,so as to achieve a much less naive augmented Lagrangian algorithm [36], for imposing a strict volume constraint.

Figure 5: (From left to right): initial, 10th and final (50th) iterations of the optimal test-case. Only the boundary∂Ω of each shape Ω is displayed.



Author's personal copy

374 C. Dapogny et al. / Journal of Computational Physics 262 (2014) 358–378

Fig. 11. (From left to right): Initial, 10th and final (50th) iterations of the optimal mast test-case. Only the boundary ∂Ω of each shape Ω is displayed on theupper range, and only the ‘interior’ part of the associated mesh TΩ of D is displayed on the corresponding cuts of the lower range.

From (10), a descent direction for J is easily revealed: displacing Ω according to the vector field

θ = τ!

Ae(uΩ ) : e(uΩ ) − ℓ"n

for a small enough fictitious time τ > 0 will yield a new shape with better performance with respect to J .We may now put this notion of shape derivative in the general framework of Section 7.1, that is, use the above algorithm

with the velocity field given by:

f (t, x,Ω) =!!

Ae(uΩ ) : e(uΩ )"(x) − ℓ

"n(x) (11)

to produce an algorithm for optimizing the shape of an initial domain Ω0 with respect to J . Note that we have been a bitelusive in writing (11) as such, for the right-hand side only stands for points x ∈ ∂Ω . This expression should actually beextended to the whole space (a tubular neighborhood of ∂Ω is enough in numerical practice); the way such an extensionshould be performed is a topic on its own in shape optimization, and we limit ourselves to referring to [27].

The proposed method is appraised on the so-called optimal mast test case, as depicted in Fig. 11: a structure, embeddedin a T-shaped box D of height 126, and width 40 at the bottom, 80 at the top, made of an isotropic elastic material of Youngmodulus E = 1 and Poisson ratio ν = 0.3, is clamped on its base, and submitted to unit vertical loads g = −ez concentratedon the left and right arms. We minimize the objective function (9) with a Lagrange multiplier ℓ = 20. We run 50 iterationsof the above algorithm; each mesh TΩn (of the whole box) has about 15 000 vertices, and the entire computation takesroughly 40 min on a laptop computer. The decrease in the objective function in the course of the optimization processis displayed in Fig. 12. A noticeable feature of the presented computation is that the topology of the evolving shape hasdramatically changed during the process, which is an inherent (and natural) achievement of the level set description of theshape evolution.

Remark 3. The shape optimization model (9) is admittedly very rough; so to speak, no volume constraint is really imposedsince it only expresses a fixed penalization of the function whose minimization is sought – namely the compliance – withthe volume function

#Ω dx (which is why the volumes of shapes change in the course of the process in the example of

Fig. 11). However, as a slight variation of this model, an update procedure of the Lagrange multiplier ℓ could be though of,so as to achieve a much less naive augmented Lagrangian algorithm [36], for imposing a strict volume constraint.

Figure 6: (From left to right): initial, 10th and final (50th) iterations of the optimal test-case. Only the ’ interior’part of the associated mesh TΩ of D is displayed on the corresponding cuts.


Section 7.4Shape optimization in fluid mechanics

Fluid mechanics

Definition 4. let Ω be a open bounded set in Rd with a smooth boundary and let u and v,two vector fields in H1(Ω,Rd).

• the stretching tensor ε of the vector field u is

ε(u) =1

2(∇u+ (∇u)t) =

(1

2

(∂ui∂xj

+∂uj

∂xi

))

1≤i,j≤d

• the doubly contracted product of two stretching tensors ε(u) and ε(v) is:

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) and u stand for the fluid pressure and the fluid velocity, respectively, atevery point of Ω, then one defines the stress tensor σ(u, p) ∈ L2(Ω, Sd(R)) by

σ(u, p) = −pId+ 2µε(u).


Fluid mechanics

Theorem 4 (Korn inequality). let Ω be an open set in Rd of class piecewise C1. Then, thereexists a constant CΩ > 0 such 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) ,

∫

Ωp dx = 0

Vdiv(Ω) =ϕ ∈ D(Ω,Rd) ,div(ϕ) = 0

V (Ω) =v ∈ H1

0(Ω,Rd) ,div(v) = 0

= Vdiv(Ω)H1

0(Ω,Rd)

Theorem 5 (De Rham lemma). let Ω be a Lipschitz bounded connected open set in Rd andlet f ∈ H−1(Ω,Rd) such that

〈f, ϕ〉H−1,H10

= 0 ∀ϕ ∈ Vdiv(Ω)

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


Fluid mechanics

The Navier-Stokes system:

− µ∆u+∇p+ (u · ∇)u = f x ∈ Ω

div(u) = 0 x ∈ Ω

u = u0 x ∈ Eu = 0 x ∈ Γ

− pn+ 2µε(u) · n = −p0n x ∈ S

where p0 ∈ R and u0 ∈ H1/2(E,Rd).

Theorem 6 (existence and uniqueness). assume that u0 ∈ H10(E,Rd) and f ∈

L3/2(Ω,Rd).There exists µ0 > 0 and ε0 > 0 such that if µ ≥ µ0 or if ‖u0‖H1

0(E) ≤ ε0 then the

Navier-Stokes problem has a unique solution (u, p) inH1(Ω)× L2(Ω).moreover, there exists C > 0 depending only on Ω such that the solution u verifies

‖u‖H10(Ω) ≤ C

(‖u0‖2H1

0(E)+ ‖u0‖H1

0(E)

)+ ‖f‖2

L3/2(Ω).


Fluid mechanics

Variational formulation:

• consider the functional spaces

W0(Ω) :=

(v, q) ∈ H10(Ω,Rd)× L2(Ω) , v = 0 on E ∪ Γ

Zu0 :=

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

endowed with the usual associated topologies.

• set p = p− p0. The variational formulation of the Navier-Stokes system writes:

find (u, p) ∈ Zu0(Ω) s.t. ∀(w,ψ) ∈W0(Ω) ,∫

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

∫

Ωψ div(u) dx = 0


A toy problem

problem and modelling

• Consider a fluid driven by the previous Navier-Stokessystem, of viscosity µ flowing inside a cannula-shapedpipe/duct. For instance, we look for the optimal shape of apipeline.

• the optimal design problem writes

infΩ∈Oad

J(Ω) where J(Ω) = 2µ∫

Ω|ε(u)|2 dx

where

• µ the viscosity of the fluid, u the velocity of the fluid at every point;

• Oad is the set of admissible shapes, for instance,E (inlet) andS (outlet) are fixed and welook for the lateral boundary such that Ω open connected subset of Rd with |Ω| = V0

(given).


A toy problem

Computation of the shape derivative

• 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 Céa’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) =∫

Ω(2µε(u) : ε(v) + [(u · ∇)u]v − pdiv(v)) dx−

∫

Ωdiv(u)q dx

+∫

Γµ · u ds


A toy problem


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

∀δp ∈ C∞0 (Ω,Rd) ,∂L∂p

(Ω, u, pv, q, ν)(δp) =∫

Ωδpdiv(v) = 0 .

and

∀δu ∈ C∞0 (Ω,Rd) ,∂L∂u

(Ω, u, pv, q, ν)(δu) = 0 .

• the last equation rewrites

2µ∫

Ωε(u) : ε(δu) dx−

∫

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

−∫

Ωdiv(δu)q dx+

∫

Γµ · δu ds = 0 .


A toy problem


• we will use the following useful integration by parts formula :

Theorem 7. let y ∈ H1(Ω,Rd) and z ∈ H2(Ω,Rd). Then,

2∫

Ωε(z) : ε(y) dx = −

∫

Ω(∆z +∇div(z)) · y dx+ 2

∫

∂Ωε(z)n · y ds .

• step 1: identification of usimilar to what was done in the elliptic case. In particular, the function ∂L

∂µ vanishes atthe saddle point yielding to u = 0 on ∂Ω.


A toy problem


• step 2: identification of the adjoint state.we are in position to identify the adjoint state:

∀δp ∈ C∞0 (Ω,Rd) ,∂L∂p

(Ω, u, pv, q, ν)(δp) =∫

Ωδpdiv(v) = 0

yields

div(v) = 0 in Ω .

using ∀δu ∈ C∞0 (Ω,Rd) ,∂L∂u

(Ω, u, pv, q, ν)(δu) = 0.

• integrating by parts this relation and making δu first with compact support in Ω andthen varying on Γ yields that (v, q) solves the following PDE:

− µ∆v + (∇u)tv − (∇v)u+∇q = −2µ∆u x ∈ Ω

div(v) = 0 x ∈ Ω

v = 0 x ∈ E ∪ Γ

− qn+ 2µε(v) · n+ (u · n)v − 4µε(u) · n = 0 x ∈ S


A toy problem


• 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 on E ∪ Γ.

• remember 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∫

Ωε(u) : ε(u′) dx+ 2µ

∫

Γ|ε(u)|2(θ · n) ds

Proposition 2. assume that Ω is of class C2. One has

J ′(Ω)(θ) = 2µ∫

Γ

(ε(u) : ε(v)− |ε(u)|2

)(θ · n) ds


A toy problem

Well posed character of the adjoint problem

• recall thatW0(Ω) =

(v, q) ∈ (H10(Ω))3 × L2(Ω) , v = 0 on E ∪ Γ

Proposition 3. The variational formulation of the adjoint state writes:

find (v, q) ∈W0(Ω) such that ∀(w,ψ) ∈W0(Ω),∫

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

∫

Ωε(u) : ε(w) dx

∫

Ω∇ψ · v dx = 0

Theorem 8 (existence and uniqueness of the adjoint state). assume d = 2,3 and Ω is ofclass C2. Introduce (v, q) the solution of the adjoint state

− µ∆v + (∇u)tv −∇vu+∇q = −2µ∆u x ∈ Ω

div(v) = 0 x ∈ Ω

v = 0 x ∈ E ∪ Γ

− qn+ 2µε(v) · n+ (u · n)v − 4µε(u) · n = 0 x ∈ SIf the viscosity µ is large enough, then the adjoint problem has a unique solution (v, q) ∈H1(Ω)× L2(Ω).


Numerical examples

• in the context of fluid mechanics (Stokes equations), one aims at minimizing the viscousdissipation 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. theminimized function is D(Ω) + `V ol(Ω), where ` is a fixed Lagrange multiplier.


Numerical examples


Numerical examples

• in the context of fluid mechanics, minimization of the viscous dissipation D(Ω) in adouble pipe.

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


Conclusion

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

• the more accurate the representation of the boundaries of shapes, the more accuratethe mechanical analyses performed on shapes (computation of uΩn , pΩn , etc...), andthe more accurate the computation of descent directions;

• ... but the more tedious and error-prone the advection step between shapes Ωn →Ωn+1.

• we have developed an open-source code in FreeFem++, which can be downloaded at theaddress: https://github.com/flomnes/optiflow

• we recommend using the latest release of FreeFem++ at http://www.freefem.org/ff++/


https://github.com/flomnes/optiflow

http://www.freefem.org/ff++/

Theoretical and Numerical Issues of Incompressible Fluid .... Frey, chap7.pdf · Theoretical and...

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Transcript of Theoretical and Numerical Issues of Incompressible Fluid .... Frey, chap7.pdf · Theoretical and...