This paper is concerned with the treatment of uncertainties in shape optimization. We consider uncertainties in the loadings, the material properties, the geometry and the vibration frequency, both in the parametric and geometric optimization setting. We minimize objective functions which are mean values, variances or failure probabilities of standard cost functions under random uncertainties. By assuming that the uncertainties are small and generated by a finite number of random variables, and using first- or second-order Taylor expansions, we propose a deterministic approach to optimize approximate objective functions. The computational cost is similar to that of a multiple load problems where the number of loads is . We demonstrate the effectiveness of our approach on various parametric and geometric optimization problems in two space dimensions.
Keywords: Shape optimization, random uncertainties, Level Set method
@article{SMAI-JCM_2015__1__83_0,
author = {Gr\'egoire Allaire and Charles Dapogny},
title = {A deterministic approximation method in shape optimization under random uncertainties},
journal = {The SMAI Journal of computational mathematics},
pages = {83--143},
publisher = {Soci\'et\'e de Math\'ematiques Appliqu\'ees et Industrielles},
volume = {1},
year = {2015},
doi = {10.5802/smai-jcm.5},
language = {en},
url = {https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.5/}
}
TY - JOUR AU - Grégoire Allaire AU - Charles Dapogny TI - A deterministic approximation method in shape optimization under random uncertainties JO - The SMAI Journal of computational mathematics PY - 2015 SP - 83 EP - 143 VL - 1 PB - Société de Mathématiques Appliquées et Industrielles UR - https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.5/ DO - 10.5802/smai-jcm.5 LA - en ID - SMAI-JCM_2015__1__83_0 ER -
%0 Journal Article %A Grégoire Allaire %A Charles Dapogny %T A deterministic approximation method in shape optimization under random uncertainties %J The SMAI Journal of computational mathematics %D 2015 %P 83-143 %V 1 %I Société de Mathématiques Appliquées et Industrielles %U https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.5/ %R 10.5802/smai-jcm.5 %G en %F SMAI-JCM_2015__1__83_0
Grégoire Allaire; Charles Dapogny. A deterministic approximation method in shape optimization under random uncertainties. The SMAI Journal of computational mathematics, Volume 1 (2015), pp. 83-143. doi : 10.5802/smai-jcm.5. https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.5/
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