Abstract
In this paper we present a methodology that allows the efficient computation of the topological derivative for semilinear elliptic problems within the averaged adjoint Lagrangian framework. The generality of our approach should also allow the extension to evolutionary and other nonlinear problems. Our strategy relies on a rescaled differential quotient of the averaged adjoint state variable which we show converges weakly to a function satisfying an equation defined in the whole space. A unique feature and advantage of this framework is that we only need to work with weakly converging subsequences of the differential quotient. This allows the computation of the topological sensitivity within a simple functional analytic framework under mild assumptions.
Highlights
Shape functions are real valued functions defined on sets of subsets of the Euclidean space Rd
The field of mathematics dealing with the minimisation of shape functions that are constrained by a partial differential equation is called PDE constrained shape optimisation
The idea of the topological derivative is to study the local behaviour of a shape function J with respect to a family of singular perturbations (Ωǫ)
Summary
Shape functions ( called shape functionals) are real valued functions defined on sets of subsets of the Euclidean space Rd. The idea of the topological derivative is to study the local behaviour of a shape function J with respect to a family of singular perturbations (Ωǫ). The topological derivative of a shape function J with respect to perturbations (Ωǫ) is defined by. A typical and general strategy to obtain the topological sensitivity is to derive the asymptotic expansion of the partial differential equation with respect to the singular perturbation of the shape [29,30]. In this paper we will show that neither the expansion (1.2) nor (1.3) are necessary to obtain the topological sensitivity for (S) For this purpose, we use a Lagrangian approach which uses the averaged adjoint variable qǫ [15, 36, 37]. We denote by Bδ(x) the ball centred at x with radius δ > 0 and set Bδ(x) := Bδ(x)
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