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HAL Id: inria-00072983

https://hal.inria.fr/inria-00072983

Submitted on 24 May 2006

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub-

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non,

Multi-Objective Optimization in CFD by Genetic Algorithms

Nathalie Marco, Jean-Antoine Desideri, Stephane Lanteri

To cite this version:

Nathalie Marco, Jean-Antoine Desideri, Stephane Lanteri. Multi-Objective Optimization in CFD by

Genetic Algorithms. RR-3686, INRIA. 1999. �inria-00072983�

ISSN 0249-6399

a p p o r t

d e r e c h e r c h e

THÈME 4

INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

Multi-Objective Optimization in CFD by Genetic Algorithms

Nathalie Marco — Jean-Antoine Désidéri — Stéphane Lanteri

N° 3686

Avril 1999

Unité de recherche INRIA Sophia Antipolis

2004, route des Lucioles, B.P. 93, 06902 Sophia Antipolis Cedex (France)

Téléphone : 04 92 38 77 77 - International : +33 4 92 38 77 77 — Fax : 04 92 38 77 65 - International : +33 4 92 38 77 65

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probability : 0.144 probability : 0.492 probability : probability :

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population Initialize

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method (niche method) dummy fitness

according to Reproduction

Crossover

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Yes

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Function f1(x,y)

25

20

15

10

5

0

4 5 2 3

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5 4 3 2 1 0

Function f2(x,y)

53qr›{sjŠ…l#™ ê ­e‹Œl.†wqr—l.€ uz ‚ qvzwXh~l ‚ l.†.qr€qry{z´€}<u†l I °

0 10 20 30 40 50 60 70 80 90 100

0 10 20 30 40 50 60 70 80

Evaluations from the 1st to the 100th generation.

53qr›{sjŠ…l ˜_š [áb—y{trs~wXqry{z:yx¨@whjl}y}jsjtvuxwXqry{z*qvzÖwhjly{­S‹Œl.†wXq©—{l€}jtvuzjl¨¯Š…y{k&whjl-Ê<Š€Œw»›{l.zjlŠXuxwqvyz

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0 2 4 6 8 10

0 2 4 6 8 10

Optimal Pareto front

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0

2

4 0

2 4

0 5 10 15

x opt Optimal solutions for f

1 (x,y) = (x−1) ² +(y−3) ²

y opt f 1 (x opt , y opt )

0

2

4 0

2 4 0

5 10 15

x opt Optimal solutions for f

2 (x,y) = (x−4) ² +(y−2.) ²

y opt f 2 (x opt , y opt )

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0 1 2 3 4 5 6 7 8 9 10

0 1 2 3 4 5 6 7

Criterion f2

Criterion f1

Pareto front without sharing

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0 2

4 6

8 10

12

0 2 4 6 8 10 12 14

2 4 6 8 10 12 14

f 1 (x,y) Evolution of the population in the criteria set

f ₂ (x,y) f 3 (x,y)

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−1 0

1 2

−2 0

2 0 2 4

x Optimal solution for f ₁

y f 1 (x,y)

−1 0

1 2

−1 0 1 2 0 2 4 6

x Optimal solutions for f ₂

y f 2 (x,y)

−2

0

2

−2 0

2 2 2.5 3 3.5 4

x Optimal solutions for f ₃

y f 3 (x,y)

0

2

4 0

5 10 2

3 4

f ₁ (x,y) Optimal Pareto front

f ₂ (x,y) f 3 (x,y)

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-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2

-0.2 0 0.2 0.4 0.6 0.8 1 1.2

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