Topological sensitivity analysis revisited for time-harmonic wave scattering problems. Part I: the free space case

What is it about?

In this paper we revisit the computation of closed-form expressions of the topological indicator function for a one step imaging algorithm of two and three-dimensional sound-soft (Dirichlet condition), sound-hard (Neumann condition), and isotropic inclusions (transmission conditions) in the free space. From the addition theorem for translated harmonics, explicit expressions of the scattered waves by infinitesimal circular (and spherical) holes subject to an incident plane wave or a compactely supported distribution of point sources are available. Then we derive the first order term in the asymptotic expansion of the Dirichlet and Neumann traces and their surface derivatives on the boundary of the singular medium perturbation. As the shape gradient of shape functionals are expressed in terms of boundary integrals involving the boundary traces of the state and the associated adjoint field, then the topological gradient formulae follow readily. We exhibit singular perturbation asymptotics that can be reused in the derivation of the topological gradient function that generates initial guesses in the iterated numerical solution of any shape optimization problem or imaging problems relying on time-harmonic acoustic waves propagation.

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