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This article is devoted to shape optimisation of contact problems in linearised elasticity, thanks to the level set method. We circumvent the shape non-differentiability, due to the contact boundary conditions, by using penalised and regularised versions of the mechanical problem. This approach is applied to five different contact models: the frictionless model, the Tresca model, the Coulomb model, the normal compliance model and the Norton-Hoff model. We consider two types of optimisation problems in our applications: first, we minimise volume under a compliance constraint, second, we optimise the normal force, with a volume constraint, which is useful to design compliant mechanisms. To illustrate the validity of the method, 2D and 3D examples are performed, the 3D examples being computed with an industrial software.
DOI : 10.5802/smai-jcm.27
Keywords: Shape and topology Optimisation; Level set method; Unilateral contact problems; Frictional contact; Penalisation and Regularisation
Maury, Aymeric 1 ; Allaire, Grégoire 2 ; Jouve, François 1
@article{SMAI-JCM_2017__3__249_0,
author = {Maury, Aymeric and Allaire, Gr\'egoire and Jouve, Fran\c{c}ois},
title = {Shape optimisation with the level set method for contact problems in linearised elasticity},
journal = {The SMAI Journal of computational mathematics},
pages = {249--292},
publisher = {Soci\'et\'e de Math\'ematiques Appliqu\'ees et Industrielles},
volume = {3},
year = {2017},
doi = {10.5802/smai-jcm.27},
zbl = {1416.74079},
mrnumber = {3722942},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.5802/smai-jcm.27/}
}
TY - JOUR AU - Maury, Aymeric AU - Allaire, Grégoire AU - Jouve, François TI - Shape optimisation with the level set method for contact problems in linearised elasticity JO - The SMAI Journal of computational mathematics PY - 2017 SP - 249 EP - 292 VL - 3 PB - Société de Mathématiques Appliquées et Industrielles UR - http://geodesic.mathdoc.fr/articles/10.5802/smai-jcm.27/ DO - 10.5802/smai-jcm.27 LA - en ID - SMAI-JCM_2017__3__249_0 ER -
%0 Journal Article %A Maury, Aymeric %A Allaire, Grégoire %A Jouve, François %T Shape optimisation with the level set method for contact problems in linearised elasticity %J The SMAI Journal of computational mathematics %D 2017 %P 249-292 %V 3 %I Société de Mathématiques Appliquées et Industrielles %U http://geodesic.mathdoc.fr/articles/10.5802/smai-jcm.27/ %R 10.5802/smai-jcm.27 %G en %F SMAI-JCM_2017__3__249_0
Maury, Aymeric; Allaire, Grégoire; Jouve, François. Shape optimisation with the level set method for contact problems in linearised elasticity. The SMAI Journal of computational mathematics, Tome 3 (2017), pp. 249-292. doi: 10.5802/smai-jcm.27
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