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Multi-phase structural optimization via a level 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. [email protected] 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 working domain. In order to optimize both the geometry and topology of the mixture we rely on the level set method for the description of the interfaces between the different phases. We discuss various approaches, based on Hadamard method of boundary variations, for computing shape derivatives which are the key ingredients for a steepest descent algorithm. The shape gradient obtained for a sharp interface involves jump of discontinuous quantities at the interface which are difficult to numerically evaluate. Therefore we suggest an alternative smoothed interface approach which yields more convenient shape derivatives. We rely on the signed distance function and we enforce a fixed width of the transition layer around the interface (a crucial property in order to avoid increasing “grey” regions of fictitious materials). It turns out that the optimization of a diffuse interface has its own interest in material science, for example to optimize functionally graded materials. Several 2-d examples of compliance minimization are numerically tested which allow us to compare the shape derivatives obtained in the sharp or smoothed interface cases.

Keywords

Shape and topology optimizationmulti-materialssigned distance function
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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