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_εA∇u_ε)+β_εu_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.