Topological sensitivity of a shape functional defined from a solution of a high order PDE
Audric Drogoul · HAL (Le Centre pour la Communication Scientifique Directe) · 2015
The topological gradient is defined as the leading term in the asymptotic expansionof 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 bySokolowski [J. Sokolowski and A. Zochowski, SIAM J. Control Optim., 37(4), pp. 1251-1272] andMasmoudi [M. Masmoudi, Computational Methods for Control Applications, vol. 16, 2001], thisnotion has been intensively developed in recent years. There are many applications such as inmechanics 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 withthe topological sensitivity of a cost function involving the m-th derivatives of a function solution of a2m order PDE’s with Neumann boundary conditions. We place us in 2D and we consider a domainperturbed by a small crack. Generally the computation of the topological gradient is known up toa 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 adjointsolution both defined on the unperturbated domain and in function of the normal of the crack. Thework is motivated by applications in edge detection (m=1 and m=3) and fine structure detection(m=2) in 2D images.