Topological sensitivity of a shape functional defined from a solution of a high order PDE - Archive ouverte HAL Accéder directement au contenu
Pré-Publication, Document De Travail Année : 2015

Topological sensitivity of a shape functional defined from a solution of a high order PDE

Audric Drogoul
  • Fonction : Auteur
  • PersonId : 772883
  • IdRef : 183105028

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.
Fichier principal
Vignette du fichier
drogoul_mLaplacian.pdf (1.17 Mo) Télécharger le fichier
Origine : Fichiers produits par l'(les) auteur(s)
Loading...

Dates et versions

hal-01097195 , version 1 (19-12-2014)
hal-01097195 , version 2 (02-07-2015)

Identifiants

  • HAL Id : hal-01097195 , version 2

Citer

Audric Drogoul. Topological sensitivity of a shape functional defined from a solution of a high order PDE. 2015. ⟨hal-01097195v2⟩
100 Consultations
70 Téléchargements

Partager

Gmail Facebook X LinkedIn More