A deterministic approximation method in shape optimization under random uncertainties
The SMAI Journal of computational mathematics, Volume 1 (2015), pp. 83-143.

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 N 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 N. We demonstrate the effectiveness of our approach on various parametric and geometric optimization problems in two space dimensions.

DOI: 10.5802/smai-jcm.5
Classification: 65C20, 65K10, 93C95
Keywords: Shape optimization, random uncertainties, Level Set method
Grégoire Allaire 1; Charles Dapogny 2

1 Centre de Mathématiques Appliquées (UMR 7641), École Polytechnique 91128 Palaiseau, France
2 Laboratoire Jean Kuntzmann, CNRS Université Joseph Fourier, Grenoble INP Université Pierre Mendès France BP 53, 38041 Grenoble Cedex 9, France
     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/}
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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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