[PPT] - An introduction to shape and topology optimization ric Bonnetier PowerPoint Presentation

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An introduction to shape and topology optimization

Éric Bonnetier∗ and Charles Dapogny†

∗ Institut Fourier, Université Grenoble-Alpes, Grenoble, France † CNRS & Laboratoire Jean Kuntzmann, Université Grenoble-Alpes, Grenoble, France

Fall, 2020

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Foreword

Shape optimization is about the minimization of an objective function J(Ω),

depending on a shape Ω of R2 or R3, under certain constraints.

Such problems have come up early in the history of sciences, and they are

ubiquitous in nature.

Nowadays, they arouse a tremendous enhusiasm in engineering.
They are at the interface between mathematics, physics, mechanical engineering

and computer science.

Shape optimization is a burning field of research!

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Part I Introduction, history and generalities about shape

ptimization

1 Some selected milestones in shape optimization

Dido’s problem and the isoperimetric inequality Shape optimization in architecture Towards “modern” shape and topology optimization

2 Generalities about shape optimization problems and examples

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Dido’s problem (I)

Dido’s problem is reported in the myth of the foundation of Carthage by

Phœnician princess Dido, in 814 B.C. (cf. Virgil’s Aeneid, ≈ 100 B.C.).

Dido fled from Tyr (actual Lebanon) after her husband got murdered by her

brother Pygmalion.

Accompanied by her fellows, she reached the Tunisian shore, where she required a

land from local king Jarbas...

... They came to this spot, where to-day you can behold the mighty

Battlements and the rising citadel of New Carthage, And purchased a site, which was named ’Bull’s Hide’ after the bargain By which they should get as much land as they could enclose with a bull’s hide... [Virgil, Aeneid]

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Dido’s problem (II)

W. Turner: “Dido Building Carthage” or “The Rise of the Carthaginian Empire” (1815).

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Dido’s problem (III)

Using modern terminology: How to surround the largest possible area A with a given contour length ℓ?

` A `0 A0

(Left) The solution to Dido’s problem in the case where the surrounded domain is limited by the sea; (right) an “unconstrained” version of Dido’s problem.

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The isoperimetric inequality (I)

Without knowing it, Queen Dido had just discovered the isoperimetric inequality:

Let Ω ⊂ R2 be a domain with “smooth enough” boundary ∂Ω. Let A be the area covered by Ω, and ℓ be the length of ∂Ω. Then, 4πA ≤ ℓ2, where equality holds if and only if Ω is a disk.

Equivalently,

Among all domains Ω ⊂ R2 with prescribed area, that with minimum perimeter is the disk.

Multiple variants of this problem exist.

Example: One may impose that the boundary of Ω should contain a non opti- mizable region (a segment).

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The isoperimetric inequality (II)

This fact was first proved in 1838 by J. Steiner, ... but the proof was false!

Actually, J. Steiner proved that, assuming that an optimal shape exist... it should then be a disk.

However, many shape optimization problems do not have a solution, for deep

mathematical and physical reasons.

Only in 1860 did K. Weierstrass complete the proof of the isoperimetric inequality

in two dimensions.

The isoperimetric inequality holds in more general contexts, for instance in three

space dimensions (H. Schwarz, 1884): Among all domains Ω ⊂ R3 with prescribed volume, that with minimum surface is the ball.

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Another occurrence of the isoperimetric inequality

Medieval cities often have a circular shape so as to minimize the perimeter of the necessary fortifications around a given population (i.e. their area).

Map of Paris during the Dark Ages.

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SLIDE 11

Part I Introduction, history and generalities about shape

ptimization

1 Some selected milestones in shape optimization

Dido’s problem and the isoperimetric inequality Shape optimization in architecture Towards “modern” shape and topology optimization

2 Generalities about shape optimization problems and examples

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The quest of architects for optimal design (I)

Structural optimization has long been a central

concern in architectural design.

One crucial step towards modern design:

the Hooke’s theorem (1675) “As hangs the flexible chain, so but inverted will stand the rigid arch.”

(Left) A chain hanging in equilibrium under the action of gravity and tension forces; (right)

an arch standing in equilibrium under gravity and compression forces.

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The quest of architects for optimal design (II)

A. Gaudi sketched the plans of the church of the Colònia Güell (1889-1914) by

relying on a funicular model so as to determine a stable assembly of columns and vaults.

(Left) Gaudi’s experimental device, (right) model of the Colònia Güell (Photo credits:

http://www.gaudidesigner.com).

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The quest of architects for optimal design (III)

Since then, optimal design concepts have attracted the attention of world-renowned architects: Heinz Isler, Gustave Eiffel, Frei Otto, etc.

They allow to model complex geometric criteria, related to the æstethics, the

constructibility, and the mechanical performance of structures.

Optimized shapes with respect to mechanical considerations have often

“elegant” outlines: their organic nature is very appreciated by architects.

(Left) A soap-film structure, coined by Frei Otto, (right) interior view of the Manheim Garden festival.

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The quest of architects for optimal design (IV)

Nowadays, modern structural optimization techniques are currently employed for

the design of large-scale buildings.

(Left) Entrance of the Qatar National Convention Center, in Doha [Sasaki et al]. (Right) Sketch of a 288m high skyscraper in Australia by Skidmore, Owings & Merrill.

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SLIDE 16

Part I Introduction, history and generalities about shape

ptimization

1 Some selected milestones in shape optimization

Dido’s problem and the isoperimetric inequality Shape optimization in architecture Towards “modern” shape and topology optimization

2 Generalities about shape optimization problems and examples

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SLIDE 17

Towards “modern” shape and topology optimization (I)

More advanced shape optimization methods

have emerged from the 1960’s, mainly due to

The

development

f

efficient numerical tools for simulating complex physical phenomena (notably the finite element method);

The increase in computational power.
One of the first fields involved is aeronautics,

where engineers were motivated to optimize airfoils so as to

Minimize the drag of aircrafts;
Increase their lift.

Sketch of the wing of an aircraft

lift drag

An airfoil subjected to the reaction of air

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Towards “modern” shape and topology optimization (II)

Concurrently, such computer-aided methods have aroused a great enthusiasm in civil and mechanical engineering.

Optimization of a torque arm (from

[KiWan])

Optimization of an arch bridge (from [ZhaMa])

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Towards “modern” shape and topology optimization (III)

Since then, much headway has been made in the mathematical and algorithmic

practice of shape and topology optimization.

Nowadays, shape and topology optimization techniques are consistently used in

industry in a wide variety of situations.

Several industrial softwares are available: OptiStruct, Ansys, Tosca, etc.

Optimization of a hip prosthesis (Photo credits: [Al]) Optimization of an automotive chassis

(from [CaBa]) 19 / 64

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Disclaimer

This course is very introductory, and by no means exhaustive, as well for

theoretical as for numerical purposes.

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

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Part I Introduction, history and generalities about shape

ptimization

1 Some selected milestones in shape optimization 2 Generalities about shape optimization problems and examples

What is a shape optimization problem? Examples of model problems

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What is a shape optimization problem? (I)

A typical shape optimization problem arises under the form:

min

Ω∈Uad

J(Ω), s.t. C(Ω) ≤ 0, where

Ω is the shape, or the design variable;
J(Ω) is an objective function to be minimized;
C(Ω) is a constraint function;
Uad is a set of admissible shapes;
In this course, the considered problems are motivated by mechanical or physical

applications; J(Ω) and C(Ω) often depend on Ω via a state uΩ, solution to a PDE posed on Ω (e.g. the linear elasticity system, or the Stokes equations).

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What is a shape optimization problem? (II)

A shape optimization process is the combination of:
A physical model, often based on PDE (e.g. the linear elasticity equations,

the Stokes system, etc...) describing the mechanical behavior of shapes,

A mathematical representation of shapes and their variations (e.g. as sets of

parameters, density functions, etc...),

A numerical description of shapes (e.g. by a mesh, a spline model, etc...)
These choices are strongly inter-dependent and they are often guided by the

particular application.

Roughly speaking, shape and topology optimization problems fall intro three main

categories: parametric, geometric and topology optimization.

This classification is quite arbitrary; it mainly reflects a point of view about what is

important in the problem. The associated mathematical and numerical methods share a lot of common features.

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Various settings for shape optimization (I)

I. Parametric optimization

The considered shapes are described by means of a set of physical parameters {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 : S → R.

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Various settings for shape optimization (II)

The parameters describing shapes are the only optimization variables, and the

shape optimization problem rewrites: min

{pi }∈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

account for variations of 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 about the sought optimal design.

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Various settings for shape optimization (III)

II. Geometric shape optimization
The topology of shapes is fixed (i.e. their num-

ber of holes in 2d).

The whole boundary ∂Ω of shapes Ω is the
ptimization variable.
Geometric optimization allows for more free-

dom than parametric optimization, since no a priori knowledge of the relevant regions of shapes to act on is required.

∂Ω Ω Optimization of Ω via “free” pertur- bations of the boundary ∂Ω.

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Various settings for shape optimization (IV)

III. Topology optimization
In many applications, the suitable topology of

shapes is unknown, and it should also be subject to optimization.

In this context, it is often preferred not to describe

the boundaries of shapes, but to resort to differ- ent representations which allow for a more natural account of topological changes. Example Describing shapes Ω as density functions h : D → [0, 1].

Ω

Optimizing a shape by acting on its topology.

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SLIDE 28

Part I Introduction, history and generalities about shape

ptimization

1 Some selected milestones in shape optimization 2 Generalities about shape optimization problems and examples

What is a shape optimization problem? Examples of model problems

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A simplified, academic example (I)

A cavity D ⊂ Rd is filled with a material with ther- mal conductivity h : D → R.

A region ΓD ⊂ ∂D is kept at temperature 0.
A heat flux g is applied on ΓN := ∂D \ ΓD.
A heat source or sink f : D → R is acting

inside D. The temperature uh : D → R within the cavity is the solution to the conductivity equation:    −div(h∇uh) = f in D, uh =

n ΓD,

h ∂uh

∂n

= g

n ΓN.

. Parametric optimization problem: the design vari- able is the conductivity distribution h ∈ Uad, where Uad = {h ∈ L∞(D), α ≤ h(x) ≤ β, x ∈ D} .

ΓD ΓN D g The considered cavity

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A simplified, academic example (II)

Examples of objective functions:

The compliance C(h) of the cavity D:

C(h) =

D

h|∇uh|2dx =

Ω

fuhdx +

ΓN

guh ds, as a measure of the heat power inside D, or of the work of the heat flux on D.

A least-square error between uh and a target temperature u0:

D(h) =

D

k(x)|uh − u0 |αdx 1

α

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

The opposite of the first eigenvalue of the cavity:

−λ1(h), where λ1(h) = min

u∈H1(D) u=0 on ΓD

D

|∇u|2 dx

Ω

u2 dx , which characterizes the decay rate of the heat inside D in the transient version of the conductivity equation.

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A simplified, academic example (III)

This problem has a geometric optimization variant, where the conductivity inside D takes

A high value β inside a region Ω ⊂ D;
A low value α inside D \ Ω;

that is: hΩ = α + χΩ(β − α), where χΩ is the characteristic function of Ω. The temperature uΩ : D → R is the solution to the conductivity equation:    −div(hΩ∇uΩ) = f in D, uΩ =

n ΓD,

hΩ

∂uΩ ∂n

= g

n ΓN.

. Geometric optimization problem: the design vari- able is the geometry Ω of the good conducting phase.

ΓD ΓN D g Ω The two-phase conductivity setting

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Shape optimization in structural mechanics (I)

We consider a structure Ω ⊂ Rd, that is, a bounded domain 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 is governed by the linear elasticity system:        −div(Ae(uΩ)) = in Ω, uΩ =

n ΓD,

Ae(uΩ)n = g

n ΓN,

Ae(uΩ)n =

n Γ := ∂Ω \ (ΓD ∪ ΓN),

where e(u) =

1 2(∇uT + ∇u) is the strain tensor field,

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

ΓD ΓN

g

A “Cantilever” beam The deformed cantilever

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Shape optimization in structure mechanics (II)

Examples of objective functions:

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

C(Ω) =

Ω

Ae(uΩ) : e(uΩ) dx =

ΓN

g.uΩ ds

A least-square discrepancy between the displacement uΩ and a target displacement

u0 (useful when designing micro-mechanisms): D(Ω) =

Ω

k(x)|uΩ − u0 |αdx 1

α

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

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SLIDE 34

Shape optimization in structure mechanics (III)

Examples of constraints:

A constraint on the volume Vol(Ω), or on the perimeter Per(Ω) of shapes.

Vol(Ω) =

Ω

dx, Per(Ω) =

∂Ω

ds.

A constraint on the total stress developped in shapes:

S(Ω) =

Ω

||σ(uΩ)||2 dx, where σ(u) = Ae(u) is the stress tensor.

Geometric constraints, e.g.
Constraints on the minimal and maximum thickness of shapes;
Constraints on their curvature radii;

such constraints are often imposed by the manufacturing process.

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SLIDE 35

Shape optimization in fluid mechanics (I)

An incompressible fluid with kinematic viscosity ν occupies 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 the Stokes equations:            −2νdiv(D(u)) + ∇p = f in Ω div(u) = 0 in Ω u = uin

n Γin

u = 0

n Γ

σ(u)n = −pout

n Γout

, where D(u) = 1

2(∇uT + ∇u) is the rate of strain tensor.

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SLIDE 36

Shape optimization in fluid mechanics (II)

Model problem I: Optimization of the shape of a pipe.

The shape is a pipe, connecting the (fixed) in-

put area Γin and output area Γout.

One is interested in minimizing the total work
f the viscous forces inside the shape:

J(Ω) = 2ν

Ω

D(uΩ) : D(uΩ) dx.

A constraint on the volume Vol(Ω) of the pipe

is enforced.

Γin Γout Γ Ω

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Shape optimization in fluid mechanics (III)

Model problem II: Reconstruction of the shape of an obstacle.

An obstacle of unknown shape ω is immersed in a fixed domain D filled by the

considered fluid.

Given a mesure umeas of the velocity uΩ of the fluid inside a small observation area

O, one aims at reconstructing the shape of ω.

The optimized domain is Ω := D \ ω, and only the part ∂ω of ∂Ω is optimized.

One then minimizes the least-square criterion: J(Ω) =

O

|uΩ − umeas|2 dx.

Ω ω Γin Γout D O

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SLIDE 38

And yet more examples

Optimization of the shape of an airfoil: reducing the drag acting on airplanes

(even by a few percents) has been a tremendous challenge in the aerodynamic industry for decades.

Optimization of the microstructure of composite materials: in linear elasticity, one

is interested in the design of negative Poisson ratio materials, etc...

Optimization of the shape of wave guides (e.g. optical fibers), in order to minimize

the power loss of conducted electromagnetic waves.

etc...

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SLIDE 39

Why are shape optimization problems difficult?

From the modeling viewpoint: difficulty to describe the physical problem at stake

by a model which is relevant (thus complicated enough), yet tractable (i.e. simple enough).

From the theoretical viewpoint: often, optimal shapes do not exist, and shape
ptimization problems enjoy at most local optima.
From both theoretical and numerical viewpoints: the optimization variable is the

domain! Hence the need for of a means to differentiate functions depending on the domain, and before that, to parametrize shapes and their variations.

On the numerical side: difficulty to represent shapes and their evolutions.
On the numerical side: shape optimization problems may be very sensitive and can

be completely dominated by discretization errors.

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SLIDE 40

Appendix: physical models

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SLIDE 41

A primer in linear elasticity

Elasticity is the ability of a structure Ω to resist an input stress, and to return exactly to its original state when the stress is relieved (= plasticity or fracture). The motion of an elastic shape Ω is described by:

The deformation ϕ : Ω → ϕ(Ω);
The displacement u(x) = ϕ(x) − x.

Ω

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u(x)

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Lagrangian point of view: the considered quantity is the motion u(x) of the constituent particles x of the structure at rest Ω, which serves as reference.

41 / 64

SLIDE 42

The strain tensor (I)

The Cauchy-Green strain tensor C(x) measures how ϕ distorts lengths. A curve (0, 1) ∋ t → γ(t) ∈ Ω with length ℓ(γ) := 1 |γ′(t)|dt; is transformed into t → ϕ(γ(t)), with length: ℓ(ϕ(γ)) = 1

(C(γ(t))γ′(t)) · γ′(t) dt,

where C(x) = (∇ϕ(x))T(∇ϕ(x)) = (I + ∇u(x))T(I + ∇u(x)).

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x

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y

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'(x)

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'(y)

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γ

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'(γ)

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'

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42 / 64

SLIDE 43

The strain tensor (II)

Geometric linearity Assuming that the displacement u is “small”, one approximates: C(x) ≈ I + ∇u(x) + ∇uT(x).

The linearized strain tensor

e(u)(x) := 1 2(∇u(x) + ∇uT(x)) then satisfies: ℓ(ϕ(γ)) ≈ ℓ(γ) + 1 e(u)(γ(t)) γ′(t) |γ′(t)| · γ′(t) |γ′(t)||γ′(t)| dt.

In coordinates, e(u) is defined by:

e(u)i,j = 1

2

∂ui

∂xj + ∂uj ∂xi

,

i, j = 1, . . . , d.

43 / 64

SLIDE 44

The stress tensor

The stress tensor σ encodes the internal efforts

within the body.

For x ∈ Ω, n ∈ Rd with |n| = 1, σ(x)n ∈ Rd is

the force applied by the surrounding medium on the face oriented by n of a small cube around x.

Cauchy’s theorem: as a consequence of balance of

momentum, σ(x) is a d × d symmetric matrix. x

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n

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σ(x)n

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e1

<latexit sha1_base64="4sztaGtETuN9eqjgtRKD62IF0=">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</latexit>

e2

<latexit sha1_base64="JBeksJf/8HiCPa7bz91uEFToh8g=">ACxnicjVHLSsNAFD2Nr1pfVZdugkVwVZJa0GXBTZcV7QNqKcl0WkPTJEwmSimCP+BWP038A/0L74xTUIvohCRnzr3nzNx7/SQMUuk4rzlraXldS2/XtjY3NreKe7utdI4E4w3WRzGouN7KQ+DiDdlIEPeSQT3Jn7I2/74XMXbt1ykQRxdyWnCexNvFAXDgHmSqEver/SLJafs6GUvAteAEsxqxMUXGOAGAwZJuCIAmH8JDS04ULBwlxPcyIE4QCHe4R4G0GWVxyvCIHdN3RLuYSPaK89UqxmdEtIrSGnjiDQx5QnC6jRbxzPtrNjfvGfaU91tSn/feE2Ilbgh9i/dPO/OlWLxBnuoaAako0o6pjxiXTXVE3t79UJckhIU7hAcUFYaV8z7bWpPq2lVvPR1/05mKVXtmcjO8q1vSgN2f41wErUrZPSlXLqlWtWMOo8DHOKY5nmKGupoEneIziCc9W3YqszLr7TLVyRrOPb8t6+ADgLpAF</latexit>

e3

<latexit sha1_base64="p4dr83jsVmxrzofXy5bkcJTFf8=">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</latexit>

σ1,1

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σ2,2

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σ3,3

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The diagonal entries of σ account for traction and compression forces.

e1

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e2

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e3

<latexit sha1_base64="p4dr83jsVmxrzofXy5bkcJTFf8=">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</latexit>

σ2,3

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σ1,2

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σ3,1

<latexit sha1_base64="ceRYeZRE82aHW6XpnoutX+m0o7c=">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</latexit>

The off-diagonal entries of σ account for shear effects.

44 / 64

SLIDE 45

The equilibrium equations

The equilibrium equations relate internal efforts with external stresses.
If body forces f : Ω → Rd (e.g. gravity) occur, it holds, for any subset V ⊂ Ω,
∂V

σ · n ds

Efforts applied on ∂V

+

V

f dx

Body efforts within V

= 0, and so, using Green’s formula:

V

(div(σ) + f ) dx = 0. Since the latter relation holds for any subset V ⊂ Ω, it follows: −div(σ) = f in Ω.

Likewise, if external forces (e.g. traction loads) g : ∂Ω → Rd are applied, it holds:

σn = g on ∂Ω.

45 / 64

SLIDE 46

The constitutive relation (I)

The system of equations is completed with a constitutive relation between the

stress tensor σ and the strain tensor e(u), which describes, equivalently,

The deformation of a piece of material caused by a given stress;
The internal stress induced by an imposed deformation.
Material linearity: σ depends linearly on e(u), via the Hooke’s law:

σ = Ae(u). e

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σ

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yield stress

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ultimate stress

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fracture

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Linear

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elasticity

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Strain

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hardening

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Necking

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46 / 64

SLIDE 47

The constitutive relation (II)

The Hooke’s law A of an isotropic material reads: ∀e ∈ Sd(R), Ae = 2µe + λtr(e)I. where the Lamé parameters λ, µ are related to the more physical quantities E and ν: µ = E 2(1 + ν), and λ = Eν (1 + ν)(1 + ν(1 − d)).

The Young’s modulus

E = σ/L measures the resistance to deformation under traction;

The Poisson’s ratio

ν = −ℓ/L accounts for the relative transverse displacement for a given longitudi- nal deformation.

σ ` L

47 / 64

SLIDE 48

Linear elasticity at a glance

In most concrete situations,

The elastic shape Ω is attached on a

region ΓD of its boundary;

Body forces f : Ω → Rd are at play;
Surface loads g : ΓN → Rd are applied
n a region ΓN ⊂ ∂Ω;
The remaining region Γ := ∂Ω \ (ΓD ∪

ΓN) of ∂Ω is traction-free.

ΓD

u(x)

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The displacement u : Ω → Rd of the shape in this context is the unique solution (in H1(Ω)d) to:        −div(Ae(u)) = f in Ω, u = 0

n ΓD,

Ae(u)n = g

n ΓN,

Ae(u)n = 0

n Γ.

48 / 64

SLIDE 49

A glimpse of incompressible fluid mechanics

A fluid body is characterized by its inability to resist a permanent shear stress.
Eulerian description: one looks at the properties inside the fluid domain Ω at all

positions x, independently of the attached particle (the latter may change).

We focus on the steady-state Stokes equations; see [ChoMar] for more advanced

models: time-dependent, Navier-Stokes equations, turbulence phenomena, etc. The state of the fluid is described in terms of

The velocity u : Ω → Rd;
The pressure p : Ω → R inside the fluid.

x

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u(x)

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p(y)

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49 / 64

SLIDE 50

Rate of strain and vorticity

Two important quantities related to the velocity of the fluid are:

The rate of strain tensor:

D(u) = 1 2(∇u + ∇uT).

The vorticity ω is a scalar field if d = 2, and a vector field if d = 3:

ω = ∂u2 ∂x1 − ∂u1 ∂x2 if d = 2; ω = ∇ × u = ∂u3 ∂x2 − ∂u2 ∂x3 , ∂u1 ∂x3 − ∂u3 ∂x1 , ∂u2 ∂x1 − ∂u1 ∂x2

if d = 3.

Physical interpretation: A Taylor expansion around x ∈ Ω yields (for d = 3): u(x + h) = u(x) + D(u)(x)h + 1 2ω(x) × h + O(|h|2);

The transformation h → D(u)(x)h induces an elongation prescribed by the

eigenvalues of the symmetric matrix D(u);

The mapping h → ω(x) × h is a rotation with axis

ω(x) |ω(x)| and velocity |ω(x)|.

50 / 64

SLIDE 51

The equilibrium equations (I)

As in the case of elasticity, the balance of momentum at equilibrium implies that:

−div(σ) = f in Ω, where:

σ : Ω → Rd×d is the stress tensor;
f : Ω → Rd represents volumic forces (e.g. gravity).
The fluid is assumed to be incompressible: at equilibrium, the mass contained

inside each subset V ⊂ Ω is conserved:

V

u · n ds = 0, and so, by virtue of Green’s formula: div(u) = 0 in Ω.

51 / 64

SLIDE 52

Newton’s law

The stress tensor σ is related to the characteristics u, p of the fluid via Newton’s law: σ = 2νD(u) − pI. Here,

The viscous forces 2νD(u) are frictional, slowering effects, proportional to the

variations of the velocity within the fluid, via the viscosity coefficient ν.

The pressure p induces a normal stress on every region of Ω.

x

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x + εe3

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u(x + εe3)

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u(x)

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σe3

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<latexit sha1_base64="TNtM6sisSmcHFa4XYth1KlNRE8=">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</latexit>

The difference between the velocity at x and x + εe3 causes a friction force at x, proportionnal to the viscosity of the fluid.

e1

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e2

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e3

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−p(x)e2

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x

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−p(x)e1

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−p(x)e3

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Pressure forces act in a normal fashion.

52 / 64

SLIDE 53

Fluid mechanics at a glance

In most practical situations,

The fluid is subjected to internal forces f : Ω → Rd;
The fluid enters the domain Ω via the region Γin ⊂ ∂Ω, with a known velocity

profile uin : Γin → Rd;

The fluid leaves Ω through the region Γout ⊂ ∂Ω, with no applied stress;
The fluid satisfies no slip boundary conditions on the remaining region

Γ := ∂Ω \ (Γin ∪ Γout), i.e. the fluid “sticks” to the wall. The Stokes equations read in this context:            −2νdiv(D(u)) + ∇p = f in Ω, div(u) = 0 in Ω, σ(u)n = 0

n Γout,

u = uin

n Γin,

u = 0

n Γ.

53 / 64

SLIDE 54

Appendix: technical facts

54 / 64

SLIDE 55

Notation for differential calculus (I)

Let Ω be an open subset of Rd;

The gradient of a (differentiable) real-valued function u : Ω → R is the vector field:

∀x ∈ Ω, ∇u(x) =    

∂u ∂x1 (x)

. . .

∂u ∂xd (x)

    .

The derivative of a vector-valued function v : Ω → R at x ∈ Ω is the (tensor)

matrix field: ∀x ∈ Ω, ∇v(x) =    

∂v1 ∂x1 (x)

. . .

∂v1 ∂xd (x)

. . . . . . . . .

∂vd ∂x1 (x)

. . .

∂vd ∂xd (x)

   

55 / 64

SLIDE 56

Notation for differential calculus (II)

The divergence of a vector field v : Ω → Rd is the function:

∀x ∈ Ω, div(v)(x) = tr(∇v(x)) = ∂v1 ∂x1 (x) + . . . + ∂vd ∂xd (x)

The divergence of a tensor field σ : Ω → Rd×d is the vector field whose entries are

the divergence of the rows of σ: ∀x ∈ Ω, div(σ)(x) =    

∂σ11 ∂x1 + . . . + ∂σ1d ∂xd

. . .

∂σd1 ∂x1 + . . . + ∂σdd ∂xd

    .

56 / 64

SLIDE 57

The Green’s formula

The Green’s formula is a generalization of integration by parts to the case of multiple space dimensions.

Theorem 1 (Green’s formula).

Let Ω ⊂ Rd be a bounded, Lipschitz domain. Then, for any function u ∈ W 1,1(Ω),

Ω

∂u ∂xi dx =

∂Ω

uni ds, i = 1, . . . , d, where n = (n1, . . . , nd) is the unit normal vector to ∂Ω, pointing outward Ω. The Green’s formula has a number of useful avatars, such as:

Corollary 2.

Let Ω ⊂ Rd be a bounded, Lipschitz domain. Then, for any u ∈ H2(Ω), v ∈ H1(Ω):

Ω

∆u v dx =

∂Ω

∂u ∂n v ds −

Ω

∇u · ∇v dx.

57 / 64

SLIDE 58

Bibliography

58 / 64

SLIDE 59

Cultural references around shape optimization I

[AllJou] G. Allaire, Design et formes optimales (I), (II) et (III), Images des Mathématiques (2009). [HilTrom] S. Hildebrandt et A. Tromba, Mathématiques et formes optimales : L’explication des structures naturelles, Pour la Science, (2009).

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SLIDE 60

General references around shape optimization I

[All] G. Allaire, Conception optimale de structures, Mathématiques & Applications, 58, Springer Verlag, Heidelberg (2006). [AlDaJou] G. Allaire, C. Dapogny and F. Jouve, Shape and topology

ptimization, to appear in Handbook of Numerical Analysis, Vol 22 – Geometric

PDEs, (2020). [BenSig] M.P. Bendsøe and O. Sigmund, Topology Optimization, Theory, Methods and Applications, 2nd Edition Springer Verlag, Berlin Heidelberg (2003). [HenPi] A. Henrot and M. Pierre, Variation et optimisation de formes, une analyse géométrique, Mathématiques et Applications 48, Springer, Heidelberg (2005). [Pironneau] O. Pironneau, Optimal Shape Design for Elliptic Systems, Springer, (1984).

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SLIDE 61

References about mechanical models I

[ChoMar] A.J. Chorin and J.E. Mardsen, A mathematical introduction to fluid mechanics, New York: Springer-Verlag, Vol. 175, (1990). [GouFen] P.L. Gould and Y. Feng, Introduction to linear elasticity, Springer, Vol. 346, (2013).

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SLIDE 62

Educational and online resources I

[Allaire2] Grégoire Allaire’s web page, http://www.cmap.polytechnique.fr/ allaire/. [Bo] B. Bogosel, Beni Bogosel’s webpage, http://www.cmap.polytechnique.fr/∼beniamin.bogosel. [AlPan] G. Allaire and O. Pantz, Structural Optimization with FreeFem++,

Struct. Multidiscip. Optim., 32, (2006), pp. 173–181.

[BonDa] E. Bonnetier and C. Dapogny, Web page of the course, https://ljk.imag.fr/membres/Charles.Dapogny/coursoptim.html. [DaFrOmPri] C. Dapogny, P. Frey, F. Omnes and Y. Privat, Geometrical shape

ptimization in Fluid Mechanics using FreeFem++, Struct. Multidisc. Optim, 58,
no. 6, (2018), pp. 2761–2788.

[DTU] Web page of the Topopt group at DTU, http://www.topopt.dtu.dk.

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SLIDE 63

Educational and online resources II

[FreyPri] P. Frey and Y. Privat, Aspects théoriques et numériques pour les fluides incompressibles - Partie II, slides of the course (in French), available at http://irma.math.unistra.fr/∼privat/cours/fluidesM2.php. [FreeFem++] Web page of the FreeFem project, https://freefem.org/. [Lau] A. Laurain, A level set-based structural optimization code using FEniCS,

Struct. Multidisc. Optim., 58, 3, (2018), pp. 1311–1334.

[Sigmund] O. Sigmund, A 99 line topology optimization code written in MATLAB, Struct. Multidiscip. Optim., 21, 2, (2001), pp. 120–127.

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SLIDE 64

Credits I

[Al] Altair hyperworks, https://insider.altairhyperworks.com. [CaBa] M. Cavazzuti, A. Baldini, E. Bertocchi, D. Costi, E. Torricelli and P. Moruzzi, High performance automotive chassis design: a topology optimization based approach, Structural and Multidisciplinary Optimization, 44, (2011),

pp. 45–56.

[KiWan] N.H. Kim, H. Wang and N.V. Queipo, Efficient Shape Optimization Under Uncertainty Using Polynomial Chaos Expansions and Local Sensitivities, AIAA Journal, 44, 5, (2006), pp. 1112–1115. [ZhaMa] X. Zhang, S. Maheshwari, A.S. Ramos Jr. and G.H. Paulino, Macroelement and Macropatch Approaches to Structural Topology Optimization Using the Ground Structure Method, Journal of Structural Engineering, 142, 11, (2016), pp. 1–14.

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An introduction to shape and topology optimization

Foreword

Contents

Part I Introduction, history and generalities about shape

Dido’s problem (I)

Dido’s problem (II)

Dido’s problem (III)

` A `0 A0

The isoperimetric inequality (I)

The isoperimetric inequality (II)

Another occurrence of the isoperimetric inequality

Part I Introduction, history and generalities about shape

The quest of architects for optimal design (I)

The quest of architects for optimal design (II)

The quest of architects for optimal design (III)

The quest of architects for optimal design (IV)

Part I Introduction, history and generalities about shape

Towards “modern” shape and topology optimization (I)

Towards “modern” shape and topology optimization (II)

Towards “modern” shape and topology optimization (III)

Disclaimer

Disclaimer

Part I Introduction, history and generalities about shape

What is a shape optimization problem? (I)

What is a shape optimization problem? (II)

Various settings for shape optimization (I)

S

h(x)

Various settings for shape optimization (II)

Various settings for shape optimization (III)

Various settings for shape optimization (IV)

Part I Introduction, history and generalities about shape

A simplified, academic example (I)

A simplified, academic example (II)

A simplified, academic example (III)

Shape optimization in structural mechanics (I)

Shape optimization in structure mechanics (II)

Shape optimization in structure mechanics (III)

Shape optimization in fluid mechanics (I)

Shape optimization in fluid mechanics (II)

Shape optimization in fluid mechanics (III)

Ω ω Γin Γout D O

And yet more examples

Why are shape optimization problems difficult?

Appendix: physical models

A primer in linear elasticity

The strain tensor (I)

The strain tensor (II)

The stress tensor

The equilibrium equations

The constitutive relation (I)

The constitutive relation (II)

σ ` L

Linear elasticity at a glance

A glimpse of incompressible fluid mechanics

Rate of strain and vorticity

The equilibrium equations (I)

Newton’s law

Fluid mechanics at a glance

Appendix: technical facts

Notation for differential calculus (I)

Notation for differential calculus (II)

The Green’s formula

Theorem 1 (Green’s formula).

Corollary 2.

Bibliography

Cultural references around shape optimization I

General references around shape optimization I

References about mechanical models I

Educational and online resources I

Educational and online resources II

Credits I