[Optimisation topologique et géométrique de structures élastiques par déformation exacte de maillage simplicial]
On présente dans cette note une méthode dʼoptimisation structurale qui sʼappuie sur deux manières complémentaires de représenter des formes : dʼune part, elles sont maillées exactement afin que lʼévaluation des performances mécaniques par éléments finis soit précise ; dʼautre part, on utilise leur représentation à lʼaide dʼune fonction de lignes de niveaux pour les déformer suivant le gradient de forme. Lʼingrédient crucial est un algorithme de remaillage qui permet de construire un maillage, de qualité appropriée pour les calculs numériques, à partir dʼune fonction ligne de niveaux continue et affine par morceaux sur un maillage non structuré. Par conséquent, notre approche peut être vue à la fois comme une méthode dʼoptimisation géométrique (puisque les structures sont maillées exactement) et comme une méthode dʼoptimisation topologique (puisque la topologie des formes successives peut changer grâce à lʼutilisation de lʼalgorithme des lignes de niveaux).
We propose a method for structural optimization that relies on two alternative descriptions of shapes: on the one hand, they are exactly meshed so that mechanical evaluations by finite elements are accurate; on the other hand, we resort to a level-set characterization to describe their deformation along the shape gradient. The key ingredient is a meshing algorithm for building a mesh, suitable for numerical computations, out of a piecewise linear level-set function on an unstructured mesh. Therefore, our approach is at the same time a geometric optimization method (since shapes are exactly meshed) and a topology optimization method (since the topology of successive shapes can change thanks to the power of the level-set method).
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@article{CRMATH_2011__349_17-18_999_0,
author = {Allaire, Gr\'egoire and Dapogny, Charles and Frey, Pascal},
title = {Topology and geometry optimization of elastic structures by exact deformation of simplicial mesh},
journal = {Comptes Rendus. Math\'ematique},
pages = {999--1003},
publisher = {Elsevier},
volume = {349},
number = {17-18},
year = {2011},
doi = {10.1016/j.crma.2011.08.012},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2011.08.012/}
}
TY - JOUR AU - Allaire, Grégoire AU - Dapogny, Charles AU - Frey, Pascal TI - Topology and geometry optimization of elastic structures by exact deformation of simplicial mesh JO - Comptes Rendus. Mathématique PY - 2011 SP - 999 EP - 1003 VL - 349 IS - 17-18 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2011.08.012/ DO - 10.1016/j.crma.2011.08.012 LA - en ID - CRMATH_2011__349_17-18_999_0 ER -
%0 Journal Article %A Allaire, Grégoire %A Dapogny, Charles %A Frey, Pascal %T Topology and geometry optimization of elastic structures by exact deformation of simplicial mesh %J Comptes Rendus. Mathématique %D 2011 %P 999-1003 %V 349 %N 17-18 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2011.08.012/ %R 10.1016/j.crma.2011.08.012 %G en %F CRMATH_2011__349_17-18_999_0
Allaire, Grégoire; Dapogny, Charles; Frey, Pascal. Topology and geometry optimization of elastic structures by exact deformation of simplicial mesh. Comptes Rendus. Mathématique, Tome 349 (2011) no. 17-18, pp. 999-1003. doi : 10.1016/j.crma.2011.08.012. http://www.numdam.org/articles/10.1016/j.crma.2011.08.012/
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