Abstract
Our main interest in this work is an analysis of geometrical inverse problem related to the detection of cavities, in elasticity framework from partially overdetermined boundary data in two spatial dimensions. For the reconstruction, we have only access to the displacement field and to the normal component of the normal stress. We propose an identification method based on the Kohn-Vogelius formulation combined with the topological gradient method. An asymptotic expansion for an energy function is derived with respect to the creation of a small hole. A one-shot reconstruction algorithm based on the topological sensitivity analysis is implemented. Some numerical experiments concerning the cavities identification are finally reported, highlighting the ability of the method to identify multiple cavities.
Highlights
Shape reconstruction of cavities is a challenging subject with application to diverse areas such as exploration geophysics, medical imaging and non-destructive testing
First case: In this case, the solution, namely the boundary of the cavity to recover, is the circle Cexact centered at the origin with radius Rexact=0.45 and the partially overdetermined boundary data, that is the displacement field and the normal component of the normal stress are taken from the analytical expression uexact, of the solution u of the elasticity problem with complete data [3], given by uexact (r)
Among several techniques proposed to solve the problem of detection of geometrical faults arising in elasticity framework, we resort, in this paper, to the topological gradient method which furnishes the sensitivity of a shape functional when modifying the topology of the domain with respect to the creation of a small hole
Summary
Shape reconstruction of cavities (namely holes) is a challenging subject with application to diverse areas such as exploration geophysics, medical imaging and non-destructive testing. In order to solve our geometrical inverse problem (1), we propose a Dirichlet-Neumann approach by the means of a self regularization technique, namely the Kohn-Vogelius formulation [3,4,7,8]. The level set method seems to be a slow and expensive process To overcome this difficulty when solving our geometrical inverse problem, we resort, in this paper, to a one iteration algorithm based on the topological gradient method. It is a recent numerical method of shape and topology optimization of structures, introduced by A.
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