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

Abstract

PurposeThe purpose of this paper is to 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).Design/methodology/approachThe 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.FindingsAs 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.Originality/valueThe authors 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 with converging Gauss−Newton iterations for the search of optimal boundary parametrizations, it generates fully automatic algorithms.

Detection of scatterers using an XFEM-BEM level set solver based on the topological derivative

“…then, a sensitivity analysis analogous to that given in [12,16] (i.e. adapting theorem 3 in [16] to the case of the domain perturbation Ω ε = Ω \ B ε (x) and using the asymptotic expressions of theorems 6 and 7 of that reference with 1/σ instead of σ due to the opposite configuration of materials in the small perturbation and the surrounding medium) shows that the expansion ( 13) is still valid, but in this case the topological derivative is given by:…”

“…The above condition has guided the development of most of the iterative reconstruction methods based on the topological derivative. If σ = 1, and assuming that the fields u Ω and w Ω are continuous across ∂Ω, then the optimality condition ( 16) implies the optimality condition ( 12), so we would expect that in this case a reconstruction satisfying (16) would have an optimal shape. However, in the case σ = 1, since the fields ∇u Ω and ∇w Ω are not continuous across ∂Ω (because of the transmission conditions), the condition ( 16) does not imply (12), and a reconstruction Ω satisfying ( 16) might not be optimal with respect to shape variations.…”

“…However, in the case σ = 1, since the fields ∇u Ω and ∇w Ω are not continuous across ∂Ω (because of the transmission conditions), the condition ( 16) does not imply (12), and a reconstruction Ω satisfying ( 16) might not be optimal with respect to shape variations. A particularly simple example was presented in [42], where it is shown that there are reconstructions that satisfy (16) which do not satisfy (12).…”

“…Then, by expressing ∇u Ω and ∇w Ω in the basis {d 1 , d 2 } and using the transmission conditions, we can easily show that the optimality condition ( 16) implies the optimality condition (12), i.e. reconstructions satisfying (16) should have an optimal shape.…”

“…The topological derivative has been used as an indicator function used to generate an image capable of revealing the location and approximate shape of the scatterers [9][10][11][12]; and thus can be used as an effective preliminary tool to create good initial guesses for gradient-based solvers. It has also been used as the basis of stand-alone iterative tools designed to obtain the final reconstruction [13][14][15][16][17].…”