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In this paper, we perform a rigourous version of shape and topologicalderivatives for optimizations problems under constraint Helmoltz problems. Ashape and topological optimization problem is formulated by introducing costfunctional. We derive first by considering the lagradian method the shapederivative of the functional. It is also proven a topological derivative withthe same approach. An application to several unconstrained shape functionsarising from differential geometry are also given.

Min max method, shape, topological derivatives, averaged Lagrangian, homogenization, two scale convergence, Helmholtz equation