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Multi-phase structural optimization via alevel set method∗∗∗

Published online by Cambridge University Press:  27 March 2014

G. Allaire ,
C. Dapogny ,
G. Delgado  and
G. Michailidis
G. Allaire
Affiliation:
CMAP, UMR 7641, Ecole Polytechnique, 91128 Palaiseau, France
C. Dapogny
Affiliation:
UPMC Univ Paris 06, UMR 7598, Laboratoire J.-L. Lions, 75005 Paris, France. dapogny@ann.jussieu.fr Renault DREAM-DELT’A, Guyancourt, France
G. Delgado
Affiliation:
CMAP, UMR 7641, Ecole Polytechnique, 91128 Palaiseau, France EADS Innovation Works, Suresnes, France
G. Michailidis
Affiliation:
CMAP, UMR 7641, Ecole Polytechnique, 91128 Palaiseau, France Renault DREAM-DELT’A, Guyancourt, France
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Abstract

We consider the optimal distribution of several elastic materials in a fixed workingdomain. In order to optimize both the geometry and topology of the mixture we rely on thelevel set method for the description of the interfaces between the different phases. Wediscuss various approaches, based on Hadamard method of boundary variations, for computingshape derivatives which are the key ingredients for a steepest descent algorithm. Theshape gradient obtained for a sharp interface involves jump of discontinuous quantities atthe interface which are difficult to numerically evaluate. Therefore we suggest analternative smoothed interface approach which yields more convenient shape derivatives. Werely on the signed distance function and we enforce a fixed width of the transition layeraround the interface (a crucial property in order to avoid increasing “grey” regions offictitious materials). It turns out that the optimization of a diffuse interface has itsown interest in material science, for example to optimize functionally graded materials.Several 2-d examples of compliance minimization are numerically tested which allow us tocompare the shape derivatives obtained in the sharp or smoothed interface cases.

Keywords

Shape and topology optimizationmulti-materialssigned distance function

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Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 20 , Issue 2 , April 2014 , pp. 576 - 611
Copyright
© EDP Sciences, SMAI, 2014

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