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Issue ESAIM: PROC
Volume 22, 2008
CANUM 2006 - Congrès National d'Analyse Numérique
Page(s) 175 - 180
DOI 10.1051/proc:072222
Published online 01 February 2008

ESAIM: Proc., 2008, Vol. 22, pp. 175-180
DOI: 10.1051/proc:072222

Multilevel gradient-based methods in aerodynamic shape design

Massimiliano Martinelli and Francois Beux

Scuola Normale Superiore di Pisa, 7 Piazza dei Cavalieri, 56126 Pisa, Italy. E-mail: max.martinelli@sns.it & fbeux@sns.it


(Published online: 1 February 2008)

Abstract
A gradient-based method coupled with a multilevel approach is proposed for shape design in aerodynamics. This method extends an existing multilevel gradient-based formulation to another type of control subspaces, i.e. considering another set of subparametrisations and prolongation operators. More precisely, Bézier control points associated with the property of degree elevation are involved instead of shape grid-points coordinates and polynomial interpolation. The good behaviour of the new formulation is demonstrated on a classical 2D nozzle inverse problem considering an adjoint formulation as well as an approximate gradient associated to a one-shot method.


Résumé
Une méthode de type gradient associée à une approche multiniveau est proposée dans le cadre de problèmes d'optimisation de forme en aérodynamique. Cette méthode généralise une formulation déjà existante de méthode de gradient multiniveau considérant un autre type de sous-espace de contrôle, c'est à dire considérant un autre ensemble de sous-paramétrisation et d'opérateur de prolongement. Plus précisement, des points de contrôle de Bézier associés à la propriété d'élévation de degré sont utilisés au lieu des coordonnées des points du maillage sur la frontière et une interpolation polynomiale. Le bon comportement de la nouvelle formulation est illustré pour le cas d'un problème inverse classique d'une tuyère bidimensionelle considérant aussi bien une méthode de l'adjoint que le calcul d'un gradient approché associé à une méthode de type "one-shot".


Mathematics Subject Classification. ???, ???

Key words: multilevel approach, optimum shape design, gradient-based method


© EDP Sciences, ESAIM 2007


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