Localization of small obstacles in Stokes flow

Abstract

We want to detect small obstacles immersed in a fluid flowing in a larger bounded domain Ω in the three dimensional case. We assume that the fluid motion is governed by the steady-state Stokes equations. We make a measurement on a part of the exterior boundary ∂Ω and then have a Kohn-Vogelius approach to locate these obstacles. We use here the notion of topological derivative in order to determine the number of objects and their rough location. Thus, we first establish an asymptotic expansion of the solution of the Stokes equations in Ω when we add small obstacles inside. Then, we use it to find a topological asymptotic expansion of the considered Kohn-Vogelius functional, which give us a formula of its topological gradient. Finally, we make some numerical simulations, exploring the efficiency and the limits of this method.

Cavities Identification from Partially Overdetermined Boundary Data in Linear Elasticity

“…As it does not impose restriction on the topology of the domain, the topological gradient method has been widely applied in literature to different and broad cases of problems as the elasticity framework [10,11], the Stokes system [12,13], the Helmholtz equation [14], the image processing problems [15,16] and many others. We refer to [13,17,18] for a concise overview of the method.…”

“…We refer to [13,17,18] for a concise overview of the method. In other words, the purpose is to study the variations of the objective function j(Ω ρ ) as ρ→0.…”

“…The computation of the topological gradient exposed below is inspired by the paper [10] and let us point out that the main contribution in this paper relies not only on the detection of cavities from partially overdetermined boundary data but also on the use of the error functional (13), also known as an energetic least-squares functional [3,4,8]. Indeed, the aim here is to derive an asymptotic expansion for the cost functional j (13) following the same procedure outlined in the previous section, that is, to study the variation of the design functional  with respect to the creation of small hole.…”

Localization of immersed obstacles from boundary measurements

“…The inverse problem consists in: given a pair of Cauchy data (g, g n ) | Γ , where g n is defined by (3), determine the location of ω. Here, we are assuming only the knowledge of the exterior boundary Γ and that at the boundary of the obstacle, γ, we have a no slip boundary condition (the geometry of γ is, therefore, unknown).…”

“…For the problem here considered we refer the work [2], where theoretical results concerning obstacle identification from boundary measurements and local stability were established. In [3] an iterative method based on the topological derivative and the Kohn-Vogelius functional is proposed for the identification of multiple 3D small obstacles and their location. In [4], a numerical shape reconstruction method based on integral equations was presented and tested.…”