Topological sensitivity of a shape functional defined from a solution of a high order PDE
Résumé
The topological gradient is defined as the leading term in the asymptotic expansion
of a shape functional with respect to the size of a local perturbation. First introduced by Schumacher
[A. Schumacher, Phd Thesis, Universitat-Gesamthochschule-Siegen, 1995] and then developped by
Sokolowski [J. Sokolowski and A. Zochowski, SIAM J. Control Optim., 37(4), pp. 1251-1272] and
Masmoudi [M. Masmoudi, Computational Methods for Control Applications, vol. 16, 2001], this
notion has been intensively developed in recent years. There are many applications such as in
mechanics of structures [S. Amstutz, I. Horchani, and M. Masmoudi, Control and Cybernetics, 34(1),
pp. 81-101, 2005], in damage evolution modelling [G. Allaire, F. Jouve, and N. Van Goethem, J.
Comput. Phys., 230(12), pp. 5010-5044, 2011] and in image processing [L. Jaafar Belaid, M. Jaoua,
M. Masmoudi, and L. Siala, Engineering Analysis with Boundary Elements, 32(11), pp. 891-899,
2008], [G. Aubert and A. Drogoul, Control, Optim. Calc. Var., to appear]. This paper deals with
the topological sensitivity of a cost function involving the m-th derivatives of a function solution of a
2m order PDE’s with Neumann boundary conditions. We place us in 2D and we consider a domain
perturbed by a small crack. Generally the computation of the topological gradient is known up to
a polarisation tensor which depends on an exterior problem and on the shape of the perturbation.
In this work we reach to fully explicit the topological gradient in function of a direct and an adjoint
solution both defined on the unperturbated domain and in function of the normal of the crack. The
work is motivated by applications in edge detection (m=1 and m=3) and fine structure detection
(m=2) in 2D images.
Origine : Fichiers produits par l'(les) auteur(s)
Loading...