Sensitivity analysis with respect to a local perturbation of the material property

In the present work, the notion of topological sensitivity is extended to the case of a local perturbation of the properties of the material constitutive of the domain. As a model example, we consider the problem −div(α ε A∇u ε )+β ε u ε =F ε in two and three dimensions, where A is a symmetric positive definite matrix and α ε ,β ε ,F ε are functions whose values inside a small subdomain ω ε are different from those of the background medium. An adjoint method is used to determine an asymptotic expansion of a given criterion when the diameter of ω ε goes to zero.