On the Ersatz material approximation in level-set methods
ESAIM: Control, Optimisation and Calculus of Variations, Volume 16 (2010) no. 3, pp. 618-634.

The level set method has become widely used in shape optimization where it allows a popular implementation of the steepest descent method. Once coupled with a ersatz material approximation [Allaire et al., J. Comput. Phys. 194 (2004) 363-393], a single mesh is only used leading to very efficient and cheap numerical schemes in optimization of structures. However, it has some limitations and cannot be applied in every situation. This work aims at exploring such a limitation. We estimate the systematic error committed by using the ersatz material approximation and, on a model case, explain that they amplifies instabilities by a second order analysis of the objective function.

DOI: 10.1051/cocv/2009023
Classification: 49Q10, 34A55, 49Q12
Keywords: shape optimization, stability, second order shape derivative, level-set method, Ersatz material approximation
     author = {Dambrine, Marc and Kateb, Djalil},
     title = {On the {Ersatz} material approximation in level-set methods},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {618--634},
     publisher = {EDP-Sciences},
     volume = {16},
     number = {3},
     year = {2010},
     doi = {10.1051/cocv/2009023},
     mrnumber = {2674629},
     zbl = {1202.49051},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/cocv/2009023/}
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JO  - ESAIM: Control, Optimisation and Calculus of Variations
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Dambrine, Marc; Kateb, Djalil. On the Ersatz material approximation in level-set methods. ESAIM: Control, Optimisation and Calculus of Variations, Volume 16 (2010) no. 3, pp. 618-634. doi : 10.1051/cocv/2009023. http://archive.numdam.org/articles/10.1051/cocv/2009023/

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