The aim of the topological sensitivity analysis is to obtain an asymptotic expansion of a shape function with respect to the variation of the topology of the structure. In the case of the creation of a small hole, it provides an
efficient method of shape optimization, but it has also successfully been
applied to shape inversion problems. In this talk, the topological asymptotic is given for the Laplace equation when a small crack is inserted
inside a plane domain.
The topological asymptotic has been numerically used to localize a family of cracks from boundary
measurements. To that end, the quadratic misfit between the ``Dirichlet'' and ``Neumann'' solutions has been used as a cost function, the topological sensitivity of which provides an accurate information regarding the
localization of the cracks. Numerical results will be presented, outlining the efficiency of the method.
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