Shape and topological sensitivity analysis in domains with cracks

Alexander Khludnev (1) , Jan Sokolowski (2) , Katarzyna Szulc (2)

1 Lavryentyev Institute of Hydrodynamics
2 IECN - Institut Élie Cartan de Nancy

Framework for shape and topology sensitivity analysis in geometrical domains with cracks is established for elastic bodies in two spatial dimensions. Equilibrium problem for elastic body with cracks is considered. Inequality type boundary conditions are prescribed at the crack faces providing a non-penetration between the crack faces. Modelling of such problems in two spatial dimensions is presented with all necessary details for further applications in shape optimization in structural mechanics. In the paper, general results on the shape and topology sensitivity analysis of this problem are provided. The results are interesting on its own. In particular, the existence of the shape and topological derivatives of the energy functional is obtained. It is shown, in fact, that the level set type method \cite{Fulman} can be applied to shape and topology opimization of the related variational inequalities for elasticity problems in domains with cracks, with the nonpenetration condition prescribed on the crack faces. The results presented in the paper can be used for numerical solution of shape optimization and inverse problems in structural mechanics.

Mots clés: Crack with non-penetration; shape sensitivity; derivative of energy functional; topological derivative

Domaines: Equations aux dérivées partielles [math.AP]

Alexander Khludnev, Jan Sokolowski, Katarzyna Szulc. Shape and topological sensitivity analysis in domains with cracks. 2008. ⟨hal-00315837⟩