Optimisation de formes par la methode

des lignes de niveau – Application a l’optimisationde composants electroniques

F. Jouve (CMAP - Ecole Polytechnique)

G. Allaire

F. De Gournay

H. Mechkour

A.M. Toader

ANR Doprocof (XLIM, Limoges) :

A. Assadi-Haghi

M. Aubourg

D. Baillargeat

S. Bila

T. Chartier

N. Delhote

S. Petit-Halajda

S. Verdeyme

www.cmap.polytechnique.fr/~optopo

Motivation

Model problem

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ω

Ω

Find the most rigid structure ω, ofprescribed volume, contained into agiven domain Ω, when given externalforces are applied.

infω⊂Ω,|ω|=V

J(ω)

where J(ω) is the objective-function.

Ill posed problem

• Not always existence of solutions (counter examples available)

• Local minima

• Numerical instabilities. No convergence under mesh refinement.

3 ways to change it into a well posed problem:

• Change the problem and enlarge the set of admissible solutions (shapes)→ relaxation (homogenization method).

• Add constraints (e.g. smoothness of the boundaries, topology) or regularizing terms(perimeter)→ narrow the set of admissible solutions

• Work on finite dimension sets and do numerical computations.→ evolutionary algorithms

1) Shape sensitivity optimization

• Hadamard method, revisited by many authors (Murat-Simon, Pironneau, Nice’sschool, etc.).

• Ill posed problem: many local minima, no convergence under mesh refinement.

• In practice, topology changes very difficult to handle.

• Very costly because of remeshing (3d).

• Very general: any constitutive equation or objective function.

• Widely used method.

2) Topology optimization

• Homogenization method (Murat-Tartar, Lurie-Cherkaev, Kohn-Strang, Bendsoe-Kikuchi, Allaire-Bonnetier-Francfort-FJ, etc.).

−→ simple objective functions, linear models but well posed problem.

• Evolutionary algorithms (Schoenauer, etc.).

• Topological asymptotics, topological gradient (Soko lowski, Masmoudi, etc.).

• Very cheap because it captures shapes on a fixed mesh.

Combine the advantages of the two approaches

• Fixed mesh → low computational cost.→ less instabilities.

• General method based on shape derivative.

Main tools:1. shape derivative (Murat-Simon),2. level set method (Osher-Sethian).

• Remaining drawbacks:1. reduction of topology (rather than variation) in 2d,2. local minima.

Hint: topological gradient

Setting of the problem

Linearly elastic material. Isotropic Hooke’s law A (but nonlinear elasticity is possible).u displacement field, e(u) = 1

2

(

∇u+ ∇uT)

deformation tensor.

Linearized elasticity system posed on ω:

−div(

Ae(u))

= 0 in ωu = 0 on ∂ω ∩ ΓD

(

Ae(u))

· n = g on ∂ω ∪ ΓN

It admits a unique solution.

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Examples of objective functions

* Compliance :

J(ω) =

∫

ω

Ae(u) : e(u)dx =

∫

ω

A−1σ : σdx =

∫

∂ω

g · u ds = c(ω)

Global measurement of rigidity.The most widely used in structural topology optimization. Good theory and very efficientnumerical methods.

* Least square criterion :

J(ω) =

∫

ω

k(x)|u(x) − u0(x)|αdx

1/α

where u0 is a given target-displacement, α ≥ 2 and k a given multiplier.

* Objective functions for compliant mechanisms optimization :

JGA(ω) = −uout

uin, JMA(ω) = −

Fout

Fin, JME(ω) = −

Foutuout

Finuin

→ MEMS design: with H.Mechkour (CEA-LIST post doctoral grant)

Numerical method strategy

1) Computation of the shape derivatives of the objectives functions (in a continuousframework). → Murat-Simon method.

2) The derivatives are discretized. The shapes are modeled by a level set function on afixed mesh. The shape is varied by advecting the level set function following the flow ofthe shape gradient.−→ transport equation of Hamilton-Jacobi type.

3) From time to time, additional computation of the topological gradient to guess whereit may be advantageous to dig new holes (and back to point 1)).

Shape derivative (Murat-Simon)

Let ω0 be a reference domain. Consider domains of the type

ω = x+ θ(x) | x ∈ ω0 =(

Id+ θ)

ω0 with θ ∈W 1,∞(IRN ; IRN).

θ(x) can be understood as a vector field advecting the reference domain ω0.

Definition:the shape derivative of J(ω) at ω0 is the Frechet differential ofθ → J

(

(Id+ θ)ω0

)

at 0.

Prop:the shape derivative J ′(ω0)(θ) depends only of θ · n on the boundary ∂ω0.

Shape derivative of the compliance

J(ω) =

∫

Γ∪ΓN

g · u ds =

∫

ω

Ae(u) · e(u) dx,

J ′(ω0)(θ) =

∫

Γ0

(

2

[

∂(g · u)

∂n+Hg · u

]

−Ae(u) · e(u)

)

θ · nds,

where u is the state (displacement field) in ω0, and H the mean curvature of Γ0.

No adjoint state involved (self-adjoint problem).

Shape derivative of the least-square criterion

J(ω) =

(∫

ω

k(x)|u− u0|αdx

)1/α

,

J ′(ω0)(θ) =

∫

Γ0

(

∂(g · p)

∂n+Hg · p−Ae(p) · e(u) +

C0

αk|u− u0|

α

)

θ · nds,

where the state u is solution of the elasticity system and the adjoint state p is solution

of the adjoint problem:

−div (Ae(p)) = C0k(x)|u− u0|α−2(u− u0) in ω0

p = 0 on ΓD(

Ae(p))

· n = 0 on ΓN ∪ Γ0,

with C0 =

(∫

ω0

k(x)|u(x) − u0(x)|αdx

)1/α−1

.

Front propagation by level set

Shapes are not meshed, but captured on a fixed mesh of a large box Ω.

Parameterization of the shape ω by a level set function:

ψ(x) = 0 ⇔ x ∈ ∂ω ∩ Ωψ(x) < 0 ⇔ x ∈ ωψ(x) > 0 ⇔ x ∈ (Ω \ ω)

• Exterior normal to ω : n = ∇ψ/|∇ψ|.• Mean curvature: H = div n.• These formula make sense everywhere in Ω, not only on the boundary ∂ω.−→ natural extension of the velocity

Hamilton-Jacobi equation

If the shape ω(t) evolves in pseudo-time t with a normal speed V (t, x), then ψsatisfies a Hamilton-Jacobi equation:

ψ(

t, x(t))

= 0 for all x(t) ∈ ∂ω(t).

deriving in t yields

∂ψ

∂t+ x(t) · ∇ψ =

∂ψ

∂t+ V n · ∇ψ = 0.

As n = ∇ψ/|∇ψ| we obtain∂ψ

∂t+ V |∇ψ| = 0.

This equation is valid on the whole domain Ω, not only on the boundary ∂ω, assumingthat the velocity is known everywhere.

Application to shape optimization

Shape derivative:

J ′(ω0)(θ) =

∫

Γ0

j(u, p, n) θ · nds.

Gradient method on the shape

ωk+1 =(

Id− j(uk, pk, nk)nk

)

ωk

because the normal nk is naturally extended to the whole Ω. The boundary of the shapeis advected with the normal speed −j.

→ Introducing a “pseudo-time” (descent parameter) we solve

∂ψ

∂t− j|∇ψ| = 0 in Ω

Numerical algorithm

1. Initialization of the level set function ψ0 (e.g. a product of sinus).

2. Iterations until convergence for k ≥ 1:

(a) Computation of uk and eventually pk by solving a linearized elasticity problem onthe shape ψk. Computation of the shape gradient → normal velocity Vk

(b) Transport of the shape by the speed Vk (Hamilton-Jacobi equation) to obtain a newshape ψk+1. (Several successive time steps can be applied for a same velocity field).The descent step is controlled by the CFL condition on the transport equation andby the decreasing of the objective function.

(c) Possible reinitialization of the level set function such that ψk+1 is the signeddistance to the interface.

3. Optionally: computation of the topological gradient to guess where holes may be dig,and return to loop 2.

Itérations

Fonc

tion-

cout

0 10 20 30 40 50 60 70 80 90 100

30

40

35

45

Xd3d 7.86 (4/11/2002)

-0.25

-0.1875

-0.125

-0.625E-01

0

0.625E-01

0.125

0.1875

0.25 et >Itération 100

-0.1

0

0.1

0.2

0.3

0

10.5

1.5

0

1

0.5

0

1

0.5

1.5

-0.1

0

0.1

0.2

0.3

0

1

0.5

Reinitialization of the level set function

To regularize the level set function and avoid it to be too flat (→ poor precision onψ) or too steep (→ poor precision on ∇ψ i.e. the normal) after some transport steps,a reinitialization is done from time to time.

We solve∂ψ

∂t+ sign(ψ)

(

|∇ψ| − 1)

= 0 in Ω,

whose stationary solution is the function signed distance to the interface:

ψ(t = 0, x) = 0

.

• Classical idea in fluid mechanics applications.• Tricky to pilot...• Reinitialization is very efficient on some problems.• Not very well understood...

Natural extension vs regularized velocity (F.de Gournay)

• Natural extension: the formula of the shape gradient is established on ∂Ω, but itcan be computed on the whole computational domain Ω.

• Hilbertian extension: obtained by solving an elliptic problem taking the velocity atthe interface ∂ω as a Neuman boundary condition.−→ the velocity is smoother−→ better convergence properties

Xd3d 8.2.3 (21 Jun 2005)

-0.4

-0.36

-0.32

-0.28

-0.24

-0.2

-0.16

-0.12

-0.8E-01

-0.4E-01

0

0.4E-01

0.8E-01

0.12

0.16

0.2

0.24

0.28

0.32

0.36

0.4 et >

Other objective-function: micromechanism design (MEMS)

input force

output force

Multiple loads MEMS

Extensions

Already done

• Multiple loads

• Design dependent loads

• Nonlinear elasticity

• Eigenvalue optimization with multiple modes

• Robust optimization

• Buckling optimization

• Stress-dependent objective functions (e.g. (∫

σα)1/α, Von Mises stresses...)

To do

• Role of the weak material

Application to electronic devices

Code coupling:

• emxd: Maxwell code developped by XLIM’s team;

• Level set engine developped at CMAP.

Interfaces:

• emxd → solution of the direct problem + adjoint state for a given geometry.

• levelset → velocity (computed using the direct solution and the adjoint state)→ new geometry.