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

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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 Lame 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 Lame 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.