Topological sensitivity analysis revisited for time-harmonic wave scattering problems. Part II: recursive computations by the boundary integral equation method

What is it about?

In this paper we revisit the recursive computation of closed-form expressions for the topological derivative of shape functionals in the context of time-harmonic acoustic waves scattering by sound-soft (Dirichlet condition), sound-hard (Neumann condition), and isotropic inclusions (transmission conditions). The elliptic boundary value problems in the singularly perturbed domains are equivalently reduced to couples of boundary integral equations with unknown densities given by boundary traces. In the case of circular or spherical holes, the spectral Fourier and Mie series expansions of the potential operators are used to derive the first order term in the asymptotic expansion of the boundary traces for the solution to the two- and three-dimensional perturbed problems. As the shape gradients 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 in the iterated numerical solution of any shape optimization or imaging problem relying on time-harmonic acoustic waves propagation. When coupled to converging Gauss-Newton iterations for the search of optimal boundary parametrizations, it generates fully automatic algorithms.

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