Conference on Applied Inverse Problems, July 20-24, 2009, Vienna, Austria
Invited Talk
Mohamed Jaoua: Detection of small flaws locations using topological asymptotic expansion

Tue, 21 July, 2009, 09:00-10:00, C1

The present work deals with the detection of small cavities in Stokes flow from over-specified
boundary data. Such a problem arises for example in moulds filling, since the industrial process
may generate small gas bubbles which are trapped inside the material while solidifying. The
inverse problem aims to locate these defects in order to decide whether the moulded piece is safe
or not. The forward problem simulation relies on quite complex and heavy models, based on the
incompressible Navier-Stokes equations in the liquid phase, and taking into account the liquid-gas
free surface as well as the solidification process.

In this work we assume that the mould is filled with a viscous incompressible fluid and we
aim to locate the unknown gas bubbles locations from boundary measurements. The velocity and
pressure of the liquid particles are governed by a simplified model based on the Stokes equations.
The gas bubbles are modelled as small cavities having an homogeneous Neumann condition on
their boundaries.

We rephrase the geometrical inverse problem under consideration into an optimal design one.
The optimal design functional to minimize in order to find out the flaws is the misfit, with
respect to some appropriate norm, between a "Dirichlet" solution based on the measurements, and
a "Neumann" one based on the prescribed loads. To minimize this misfit functional we resort the
topological gradient method. It consists in studying the sensitivity of the cost function with
respect to a small topological perturbation of the domain.

In the theoretical part, we derive a topological sensitivity analysis for the Stokes system with
respect to the insertion of a small hole (flaw) in the fluid flow domain with Neumann condition
on the boundary. The obtained results are general and valid for a large class of cost function.
The topological sensitivity of the misfit functional with respect to the presence of a "small flaw"
is computed, and it turns out to rely only on quantities needing to be computed on the safe domain.

In the numerical part, we propose a simple, fast and accurate identification procedure. The flaws
location are obtained as the most negative local minima of the misfit functional sensitivity. The
efficiency of the proposed method is illustrated by several numerical experiments. The sensitivity
of the proposed method to some numerical parameters or practical possibly occurring situations
such as the relative mesh/flaw size, the flaw's depth, the noisy data and the multi-flaws situations
are discussed.

Presentation slides (pdf, 4.6 MB)

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