The aim of this work is an analysis of some geometrical inverse problems related to the identification of cavities in linear elasticity framework. We rephrase the inverse problem into a shape optimization one using an energetic least-squares functional. The shape derivative of this cost functional is combined with the level set method in a steepest descent algorithm to solve the shape optimization problem. The efficiency of this approach is illustrated by several numerical results.
This work is devoted to some geometric inverse problems in linear elasticity. The problem considered is the cavities identification in mechanical structures from the knowledge of partially overdetermined boundary data, namely the displacement field and the normal component of the normal stress. We state a uniqueness result from a single pair of data under some geometrical assumptions. We propose an iterative method based on the coupling of the data completion process through the Steklov-Poincaré operator to reconstruct the shear stress and of the shape gradient method combined with the level set method to identify cavities. Numerical simulations highlight the algorithm efficiency.
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