The moulds' filling process may generate flaws consisting of small gas bubbles trapped inside the material, which weaken the solidity of the casted piece. We consider here the inverse problem of determining these small size flaws' locations from velocities boundary measurements. The fluid flow is described by a simplified model based on the Stokes system. A numerical algorithm based on the topological sensitivity analysis applied to an energy-like misfit functional is worked out to that end.
"We make a boundary measurement on the part of the exterior boundary ∂Ω and then study a Kohn-Vogelius type cost functional. The topological asymptotic expansion of this kind of functional has studied by Ben Abda et al. in  but they impose Neumann boundary conditions on the boundary of the objects. Here, we have to deal with Dirichlet boundary conditions on the inclusions boundaries and the Kohn-Vogelius approach leads to consider Dirichlet and mixed boundary conditions on the exterior boundary. "
[Show abstract][Hide abstract] 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 take a Kohn–Vogelius approach to locate these obstacles. We use here the notion of the topological derivative in order to determine the number of objects and their rough locations. 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 gives us the formula of its topological gradient. Finally, we make some numerical simulations exploring the efficiency and the limits of this method.
"Its aim is twofold: (i) to establish the expressions of coefficients T 2 , T 3 , T 4 for a crack of size a embedded in a 2-D medium characterized by a scalar conductivity, permitting computationally efficient methods for evaluating small-crack expansions of cost functionals, and (ii) to demonstrate the effectiveness of the resulting expansion (1) for crack identification purposes. Since the sensitivity of cost functionals (rather than field variables ) is emphasized here, an adjoint solution-based approach is formulated as it avoids the (involved and costly) evaluation of higher-order sensitivities of field variables, following what is now common practice in usual sensitivity analyses and previous works on topological sensitivity      . Coefficients T 2 , T 3 , T 4 are hence found to be expressed in terms of the free and adjoint fields (i.e. the response of the reference medium to the applied and adjoint excitations), and also (for T 4 ) on the Green's function associated with the geometry and boundary condition structure under consideration. "
[Show abstract][Hide abstract] ABSTRACT: This article concerns an extension of the topological sensitivity (TS) concept for 2D potential problems involving insulated cracks, whereby a misfit functional J is expanded in powers of the characteristic size a of a crack. Going beyond the standard TS, which evaluates (in the present context) the leading O(a2) approximation of J, the higher-order TS established here for a small crack of arbitrarily given location and shape embedded in a 2-D region of arbitrary shape and conductivity yields the O(a4) approximation of J. Simpler and more explicit versions of this formulation are obtained for a centrally symmetric crack and a straight crack. A simple approximate global procedure for crack identification, based on minimizing the O(a4) expansion of J over a dense search grid, is proposed and demonstrated on a synthetic numerical example. BIE formulations are prominently used in both the mathematical treatment leading to the O(a4) approximation of J and the subsequent numerical experiments.
Engineering Analysis with Boundary Elements 02/2011; 35(2-35):223-235. DOI:10.1016/j.enganabound.2010.08.007 · 1.39 Impact Factor
"Like in earlier works such as     , a heuristic identification method based on the computation of the TS field T (z) consists in considering locations where T (z) attains its lowest negative values as the most likely sites for a defect, as T (z) < 0 indicates that the cost functional will decrease under the nucleation of a sufficiently small inclusion at z. To focus on areas of Ω where T attains sufficiently low (negative) values, a thresholded version T α of T depending on a cut-off parameter α is defined (with the "
[Show abstract][Hide abstract] ABSTRACT: This article is concerned with establishing the topological sensitivity (TS) against the nucleation of small trial inclusions of an energy-like cost function. The latter measures the discrepancy between two time-harmonic elastodynamic states (respectively defined, for cases where overdetermined boundary data is available for identification purposes, in terms of Dirichlet or Neumann boundary data for the same reference solid) as the strain energy of their difference. Such cost function constitutes a particular form of error in constitutive relation and may be used for e.g. defect identification. The TS is expressed in terms of four elastodynamic fields, namely the free and adjoint solutions for Dirichlet or Neumann data. A similar result is also given for the linear acoustic scalar case. A synthetic numerical example where the TS result is used for the qualitative identification of an inclusion is presented for a simple 2D acoustic configuration.
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