Topological sensitivity analysis and Kohn-Vogelius formulation for detecting a rigid inclusion in an elastic body (Q1988615)

The author explores a now classical method to detect a rigid inclusion immersed in an isotropic elastic body from a single pair of Cauchy data on the boundary of a two-dimensional domain \(\Omega\). To solve this inverse problem, the author introduces a functional à la Kohn-Vogelius, which measures the difference between the solutions of two auxiliary problems. He also computes the so-called topological derivative and propose a non-iterative reconstruction algorithm based on this derivative. More precisely, the aim is to reconstruct the location and shape of a rigid inclusion \(\omega\), with Dirichlet boundary condition on its boundary thanks to over-determined data (given traction and measured displacement field) on the boundary of \(\Omega\). The state equation is the classical linear elasticity with a given traction acting on the boundary \(\partial\Omega\). The theoretical part consists in computing rigorously the topological derivative of the functional using precise asymptotic analysis. In the numerical algorithm, the unknown rigid inclusion is defined by a level curve of a scalar function. The efficiency of the proposed algorithm is illustrated by some numerical examples.