Multi-phase structural optimization via a level set method

G. Allaire; C. Dapogny; G. Delgado; G. Michailidis

ESAIM: Control, Optimisation and Calculus of Variations (2014)

  • Volume: 20, Issue: 2, page 576-611
  • ISSN: 1292-8119

Abstract

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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.

How to cite

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Allaire, G., et al. "Multi-phase structural optimization via a level set method." ESAIM: Control, Optimisation and Calculus of Variations 20.2 (2014): 576-611. <http://eudml.org/doc/272828>.

@article{Allaire2014,
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.},
author = {Allaire, G., Dapogny, C., Delgado, G., Michailidis, G.},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {shape and topology optimization; multi-materials; signed distance function; shape optimization; topology optimization},
language = {eng},
number = {2},
pages = {576-611},
publisher = {EDP-Sciences},
title = {Multi-phase structural optimization via a level set method},
url = {http://eudml.org/doc/272828},
volume = {20},
year = {2014},
}

TY - JOUR
AU - Allaire, G.
AU - Dapogny, C.
AU - Delgado, G.
AU - Michailidis, G.
TI - Multi-phase structural optimization via a level set method
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2014
PB - EDP-Sciences
VL - 20
IS - 2
SP - 576
EP - 611
AB - 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.
LA - eng
KW - shape and topology optimization; multi-materials; signed distance function; shape optimization; topology optimization
UR - http://eudml.org/doc/272828
ER -

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