Skip to main content

A first order approach for worst-case shape optimization of the compliance for a mixture in the low contrast regime


The purpose of this article is to propose a deterministic method for optimizing a structure considering its worst possible behaviour when a small uncertainty exists over its Lamé parameters. The idea is to take advantage of the small parameter to derive an asymptotic expansion of the displacement and of the compliance with respect to the contrast in Lamé coefficients. We are then able to compute the worst case design as post-treatment of the computation of the displacement field for the nominal parameters. The domain evolution resulting from the optimization is performed here using the level set method. The computational cost of our method remains of the same order as the cost of the optimization for a homogeneous material.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16


Download references


The authors thank the anonymous referees for their careful reading and helpful comments which helped improve the paper. Marc Dambrine acknowledges the support of the ANR through the grants ARAMIS-12-BS01-0021 and OPTIFORM-12-BS01-0007-02. Antoine Laurain acknowledges financial support from the DFG Research Center MATHEON “Mathematics for key technologies” through the MATHEON-Project C37 “Shape / Topology optimization methods for inverse problems”, under which part of this research has been conducted.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Marc Dambrine.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Dambrine, M., Laurain, A. A first order approach for worst-case shape optimization of the compliance for a mixture in the low contrast regime. Struct Multidisc Optim 54, 215–231 (2016).

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


  • Robust shape optimization
  • Asymptotic analysis
  • Uncertainties on material coefficients