d´eveloppement r´ecents et perspectives en optimisation de

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Developpement recents et perspectives

en optimisation de forme

et applications a la fabrication additive

Gregoire ALLAIRE, et de nombreux collaborateurs...

Post-docs: Beniamin BOGOSEL, Lukas JAKABCIN

Doctorants: Mathilde BOISSIER, Florian FEPPON, Perle

GEOFFROY, Lalaina RAKOTONDRAINIBE

Journee de rentree du CMAP, 4 octobre 2017.

Optimisation topologique et fabrication additive G. Allaire

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PLAN DE L’EXPOSE

1. Introduction

Fabrication additive.

Rappel sur la methode des lignes de niveaux pour l’optimisation

geometrique et topologique.

2. Contraintes de fabrication pour la fonderie.

3. Contraintes pour la fabrication additive.

Limitation des porte-a-faux.

Creation de supports.

Minimisation des contraintes thermiques residuelles.


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Equipe Mecanique, materiaux et optimisation de formes

Gregoire Allaire

Robert Brizzi

Antonin Chambolle

Vincent Giovangigli

Nicole Spillane

Thomas Wick

Post-docs: Beniamin BOGOSEL, Lukas JAKABCIN

Doctorants: Alois BISSUEL (Dassault Aviation), Mathilde BOISSIER

(CNRS), Qingqing FENG (CEA), Florian FEPPON (Safran), Perle

GEOFFROY (Safran), Sofiane HOUBAR (CEA), Lalaina

RAKOTONDRAINIBE (Renault)


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-I- INTRODUCTION

Additive manufacturing: structures built layer by layer.

Metallic powder melted by a laser or an electron beam.


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Shape and topology optimization

Tremendous progresses were achieved on academic research about shape

and topology optimization.

Many commercial softwares which are heavily used by industry.

Pending issue: manufacturability.


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Shape and topology optimization

Minimize an objective function J(Ω) over a set Uad of admissibles shapes Ω

(including possible topology changes)

infΩ∈Uad,P (Ω)≤0

J(Ω)

with a (possible) constraint P (Ω)

J(Ω) =

∫

Ω

j(uΩ) dx , P (Ω) =

∫

Ω

c(uΩ) dx

where uΩ is the solution of a partial differential equation (state equation)

PDE(uΩ) = 0 in Ω


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The model of linearized elasticity

For a given applied load g : ΓN → Rd, the displacement u : Ω → R

d is the

solution of

− div (Ae(u)) = 0 in Ω

u = 0 on ΓD(

Ae(u))

n = g on ΓN(

Ae(u))

n = 0 on Γ

with the strain tensor e(u) = 12

(

∇u+ (∇u)T)

, the stress tensor σ = Ae(u),

and A an homogeneous isotropic elasticity tensor.

Typical objective function: compliance

J(Ω) =

∫

ΓN

g · u dx,

Typical constraint: fixed volume or weight.


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LEVEL SET METHOD

Framework of Hadamard’s method of shape variations.

Main tool: the level set method of Osher and Sethian (JCP 1988).

Early references: Sethian and Wiegmann (JCP 2000), Osher and Santosa

(JCP 2001), Allaire, Jouve and Toader (CRAS 2002, JCP 2004, CMAME

2005), Wang, Wang and Guo (CMAME 2003).


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FRONT PROPAGATION BY LEVEL SET

Shape capturing method on a fixed mesh of the “working domain” D.

A shape Ω is parametrized by a level set function

ψ(x) = 0 ⇔ x ∈ ∂Ω ∩D

ψ(x) < 0 ⇔ x ∈ Ω

ψ(x) > 0 ⇔ x ∈ (D \ Ω)

Assume that the shape Ω(t) evolves in time t with a normal velocity V (t, x).

Then its motion is governed by the following Hamilton Jacobi equation

∂ψ

∂t+ V |∇xψ| = 0 in D.


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Advection velocity = shape gradient

The velocity V is deduced from the shape gradient of the objective function.

To compute this shape gradient we recall Hadamard’s method.

Let Ω0 be a reference domain. Shapes are parametrized by a vector field θ

Ω = ( Id + θ)Ω0 with θ ∈ C1(Rd;Rd).

x

Ω

x+ (x)θ

0 d 0(Ι +θ)Ω


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Shape derivative

Definition: the shape derivative of J(Ω) at Ω0 is the Frechet differential of

θ → J(

( Id + θ)Ω0

)

at 0.

Hadamard structure theorem: the shape derivative of J(Ω) can always be

written (in a distributional sense)

J ′(Ω0)(θ) =

∫

∂Ω0

θ(x) · n(x) j(x) ds

where j(x) is an integrand depending on the state u and an adjoint p.

We choose the velocity V = θ · n such that J ′(Ω0)(θ) ≤ 0.

Simplest choice: V = θ · n = −j but other ones are possible (including

regularization).


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NUMERICAL ALGORITHM

1. Initialization of the level set function ψ0 (including holes).

2. Iteration until convergence for k ≥ 1:

(a) Compute the elastic displacement uk for the shape ψk.

Deduce the shape gradient = normal velocity = Vk

(b) Advect the shape with Vk (solving the Hamilton Jacobi equation) to

obtain a new shape ψk+1.

——————————————————————————————

For numerical examples, see the web page:

http://www.cmap.polytechnique.fr/˜optopo/level en.html

Optimization algorithms:

1. Lagrangian (possibly augmented) algorithm,

2. SLP (sequential linear programming).


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Examples of results with complex topologies

Hard to manufacture with traditional technologies (e.g. casting).

Either, add geometrical constraints.

Or, go for additive manufacturing.


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-II- Contraintes de fabrication pour la fonderie

Avec Francois Jouve et Georgios Michailidis (projet Rodin 2012-2015)

G. Allaire, F. Jouve, G. Michailidis, Thickness control in structural

optimization via a level set method, SMO 53, 1349-1382 (2016).


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Classical geometric constraints

The signed-distance function gives a clear definition of maximal and minimal

thickness (as well as a clear penalization of draw direction).

Maximum or minimum thickness.

Molding constraints.


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Maximum thickness (3d Box)


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Minimum thickness (3d)


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No constraint (top), vertical draw direction (bottom).

Parting surface fixed at bottom (left) and free (right).


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-III- Contraintes pour la fabrication additive

Projet SOFIA (SOlutions pour la Fabrication Industrielle Additive

metallique) avec Add-Up, Michelin, Safran, ESI, etc.

Although there are less constraints than for casting, here is a partial list of

constraints for additive manufacturing:

preferred orientation of thin and slender structures,

overhang limitation,

thermal residual stress,

minimal time (or energy) for completion,

removing the powder (no closed holes).

The previous geometrical constraints may be useful but new ones are required.


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Overhang limitation

Example of a bad 3-d printing due to overhangs.

En collaboration avec Charles Dapogny, Rafael Estevez, Alexis Faure et

Georgios Michailidis


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Geometric constraint on the normal angle

To avoid bad 3-d printing due to overhangs, we forbid some angles of the

normal to the shape with the build direction d.

For a given angle φ, our pointwise criterion reads

n(x) · d ≤ cosφ, ∀x ∈ ∂Ω.

Denoting (·)+ ≡ max(·, 0), a global penalized formulation is

P (Ω) =

∫

∂Ω

[

(

n(s) · d− cosφ)+

]2

ds


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Geometric constraint

Initialization (left) and optimal designs: without constraint (middle) and with

constraint (right).

Vertical angle between −45 and 45.


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Dripping effect


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Explanation of the dripping effect

Iterations

Pnd

10 205 150.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

Horizontal boundaries are replaced by oscillating boundaries.


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New mechanical constraint for overhang limitation

Define intermediate ”layer by layer” shapes Ωi, 1 ≤ i ≤ n.

Apply self-weight to the shapes Ωi, compute its compliance, sum them up and

apply an upper bound constraint.


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For each shape Ωi, ui solves the elasticity system:

− div (Ae(ui)) = ρg in Ωi,

ui = 0 on ΓD,(

Ae(ui))

n = 0 on Γi,

Global self-weight compliance constraint:

P (Ω) =

n∑

i=1

∫

Ωi

Ae(ui) : e(ui) dx =

n∑

i=1

∫

Ωi

ρg · ui dx


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Self-weight compliance constraint

We solve the optimization problem:

minΩ⊂D

Compliance

s.t. V (Ω) ≤ 0.20|D|

P (Ω) ≤ αP (Ωref ), α ∈ (0, 1).

where Ωref is the optimal design without constraint and α is a parameter of

the method.


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Optimal designs: without constraint (left), decreasing parameter

α = 0.8, 0.5, 0.3 (right).


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Upper-weight compliance constraint

Variant for a better efficiency: only the upper part of the structure is

loaded.

gδ(x) =

g if hi − δ < xd < hi,

0 otherwise,

where hi is the height of Ωi and δ > 0.

For each shape Ωi, ui solves the elasticity system:

− div (Ae(ui)) = ρgδ in Ωi,

ui = 0 on ΓD,(

Ae(ui))

n = 0 on Γi,

Global upper-weight compliance constraint:

P (Ω) =n∑

i=1

∫

Ωi

Ae(ui) : e(ui) dx =n∑

i=1

∫

Ωi

ρgδ · ui dx


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Upper-weight compliance constraint


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Upper-weight compliance constraint in 3-d


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Mixing geometric and mechanical criteria


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Ajout de supports optimises

Travail de Beniamin Bogosel.

Modele simplifie:

1. On garde la forme finale fixe.

2. On ajoute des supports de volume minimal et permettant a la forme de

”tenir” sous l’effet de la gravite.


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Contraintes thermiques residuelles

Travail de Lukas Jakabcin.

Modele plus precis (mais plus cher):

Keep the intermediate ”layer by layer” shapes Ωi, 1 ≤ i ≤ n.

Each layer i is built between time ti−1 and ti.

Holes are now filled by a metallic powder.

Thermal residual stress computed by a model introduced in

L. Van Belle, J.-C. Boyer, G. Vansteenkiste,

Investigation of residual stresses induced during the selective laser melting

process, Key Engineering Materials, 1828-2834 (2013).


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Thermo-mechanical model

Heat equation:

ρ∂T

∂t− div(λ∇T ) = Q(t) in (ti−1, ti)× Ωi

Thermoelastic equilibrium equation:

− div(σ) = ρg and σ = σel + σth in (ti−1, ti)× Ωi,

σel = A(e(u)− p) and σth = K(T − Tinit)In,

Plastic flow rule:

∂p

∂t= (σ − PK(σ)) /β in (ti−1, ti)× Ωi

where β > 0 is the Perzyna regularization and PK is the projection on the

convex set K given, in the case of the Von Mises criterion, by

PK(σ) = σ −max(0, 1−σc|σD|

)σD

with σD the deviatoric part of the stress σ and σc > 0 the plastic yield limit.


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Thermo-mechanical constraint

From this we build an objective (or constraint) function which is

J(Ω) =

∫

Ω

j(ufinal) dx+n∑

i=1

∫ ti

ti−1

∫

Ωi

k(u, T, p) dx dt

where ufinal is the elastic displacement for the final shape, and (u, T, p) is

the displacement, temperature and plastic field for the intermediate shapes.

Steady linearized elasticity for the final shape

− div (Ae(ufinal)) = f in Ω

We compute the shape derivative of J(Ω) by an adjoint method.

The adjoints are backward in time (huge cost and storage !).

The shape boundary ∂Ω is an interface (between metal and powder) for

the intermediate shapes.

The heat source Q(t) (laser beam) is travelling in time.


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Work still going on...

For the moment, 2-d computations with FreeFem++.

First computations without plasticity.

How to choose the thermo-mechanic objective function for maximal

efficiency ?


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G. Allaire, Ch. Dapogny, A. Faure, G. Michailidis, Shape optimization of a

layer by layer mechanical constraint for additive manufacturing, C. R. Math.

Acad. Sci. Paris, 355, no. 6, 699-717 (2017). HAL preprint: hal-01398877.

G. Allaire, C. Dapogny, R. Estevez, A. Faure and G. Michailidis, Structural

optimization under overhang constraints imposed by additive manufacturing

technologies, to appear in J. Comput. Phys. HAL preprint hal-01523009 (May

2017).

Work still going on...


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