On the robustness of the topological derivative for Helmholtz problems and applications

We consider Helmholtz problems in two and three dimensions. The topological sensitivity of a given cost function J(u_∈) with respect to a small hole B_∈ around a given point x₀ ∈ B_∈ ⊂ Ω depends on various parameters, like the frequency k chosen or certain material parameters or even the shape parameters of the hole B_∈. These parameters are either deliberately chosen in a certain range, as, e.g., the frequencies, or are known only up to some bounds. The problem arises as to whether one can obtain a uniform design using the topological gradient. We show that for 2-d and 3-d Helmholtz problems such a robust design is achievable.