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The extended adjoint method

Published online by Cambridge University Press:  31 July 2012

Stanislas Larnier  and
Mohamed Masmoudi
Stanislas Larnier
Affiliation:
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
Mohamed Masmoudi
Affiliation:
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
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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.

Keywords

Adjoint methodtopology optimizationcalculus of variations
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 47 , Issue 1 , January 2013 , pp. 83 - 108
Copyright
© EDP Sciences, SMAI, 2012

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