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SENSITIVITY ANALYSIS WITH RESPECT TO A LOCAL PERTURBATION OF THE MATERIAL PROPERTY

17 pages
SENSITIVITY ANALYSIS WITH RESPECT TO A LOCAL PERTURBATION OF THE MATERIAL PROPERTY SAMUEL AMSTUTZ Abstract. 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 (??A?u?) + ??u? = F? in two and three dimensions, where A is a symmetric positive definite matrix and ??, ??, F? are functions whose values inside a small subdomain ?? 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 ?? goes to zero. 1. Introduction In the last few years, the notion of topological sensitivity has become increasingly widespread in the shape optimization community. In contrast to the classical techniques of boundary vari- ation, this tool, among some others like homogenization or level-set based methods, allows to deal with problems for which the topology (i.e. the number of holes) of the optimal domain is a priori unknown. The principle consists in studying directly the behavior of the shape functional of interest when creating a small hole inside the domain. From the mathematical point of view, given a criterion J (?), ? ? Rd (d=2 or 3), a point x0 ? ? and a fixed domain ? ? Rd, one searches for an asymptotic expansion of the form J (?\(x0 + ??))? J (?) = f(?)g

  • let ?

  • p?1 ?

  • dirichlet condition

  • boundary ∂?

  • adjoint state

  • topological sensitivity

  • parameter ?

  • ?0

  • ∂?


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SENSITIVITY ANALYSIS WITH RESPECT TO A LOCAL PERTURBATION OF THE MATERIAL PROPERTY
SAMUEL AMSTUTZ
Abstract. 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.
1. Introduction In the last few years, the notion of topological sensitivity has become increasingly widespread in the shape optimization community. In contrast to the classical techniques of boundary vari-ation, this tool, among some others like homogenization or level-set based methods, allows to deal with problems for which the topology ( i.e. the number of holes) of the optimal domain is a priori unknown. The principle consists in studying directly the behavior of the shape functional of interest when creating a small hole inside the domain. From the mathematical point of view, given a criterion J (Ω), Ω R d (d=2 or 3), a point x 0 Ω and a fixed domain ω R d , one searches for an asymptotic expansion of the form J \ ( x 0 + εω )) − J (Ω) = f ( ε ) g ( x 0 ) + o ( f ( ε )) (1.1) where f ( ε ) is an explicit positive function going to zero with ε . The function g is commonly called “topological gradient” or “topological derivative”, and (1.1) the “topological asymptotic expansion”. Therefore, to minimize the criterion J , one has to create holes at some points where the topological gradient is negative. This approach was instigated by Schumacher et al. [22], and then developed by many authors. For more details about the mathematical aspects and the related numerical procedures, the reader is referred e.g. to the publications [23, 16, 13, 10, 14, 18, 8]. Another situation, firstly addressed by Cedio-Fengya et al. [11], consists in studying the influence of the insertion of a small inhomogeneity which is nonempty, but whose constitutive parameters are different from those of the background medium. Other references on this topic can be found e.g. in [5, 4, 3, 2, 7, 1]. These works present two major differences with the previous ones. First, an interface condition holds on the border of the inclusion, instead of a classical boundary condition (usually of Dirichlet or Neumann type). Second, the authors being merely concerned by identification problems by means of boundary measurements, they provide asymptotic formulas either for objective functions specifically designed to this purpose, or of the solution itself at the location of the sensors. Therefore an adjoint state is not (at least explicitely) involved. In the present work, the link between the two above approaches is investigated. A Helmholtz type state equation with an inhomogeneity in the coefficients is considered. For a certain class of objective functions, an asymptotic expansion of the form (1.1) is derived with the help of 1991 Mathematics Subject Classification. 35J05,49N45,49Q10,49Q12. Key words and phrases. topology optimization, topological sensitivity analysis, detection of inclusions. 1