The extended adjoint method

UniversitéPaul Sabatier, Institut de Mathématiques de Toulouse, 118 route de Narbonne, 31062 Toulouse, France
stanislas.larnier@math.univ-toulouse.fr; mohamed.masmoudi@math.univ-toulouse.fr

Received: 18 November 2011
Revised: 18 April 2012

Abstract

Searching for the optimal partitioning of a domain leads to the use of the adjoint method in topological asymptotic expansions to know the influence of a domain perturbation on a cost function. Our approach works by restricting to local subproblems containing the perturbation and outperforms the adjoint method by providing approximations of higher order. It is a universal tool, easily adapted to different kinds of real problems and does not need the fundamental solution of the problem; furthermore our approach allows to consider finite perturbations and not infinitesimal ones. This paper provides theoretical justifications in the linear case and presents some applications with topological perturbations, continuous perturbations and mesh perturbations. This proposed approach can also be used to update the solution of singularly perturbed problems.