Topological asymptotic expansions for the generalized Poisson problem with small inclusions and applications in lubrication

Abstract

The asymptotic expansion of the solution u to the equation ∇ ⋅ (α∇u) = −∇ ⋅ β + γ in , with respect to the size ϵ of an inclusion (at a point x0 of the domain Ω) in which the parameters α, β and γ are changed, is studied. Writing the difference with the unperturbed solution as u − uϵ = ϵnzϵ, it is shown that the sequence zϵ converges weakly, for all p < n/(n − 1) and ρ > 0, to a function z in Lp(Ω)∩H1(Ω ∖ Bρ(x0)), where Bρ(x0) is the ball of radius ρ around x0. This allows for the calculation of asymptotic expansions of cost functions of the form J(ϵ) = ∫ΩF(uϵ) dx, for example, rendering it useful for many applications. It also extends available estimates which hold uniformly on the boundary of Ω. In addition, a link is provided with the adjoint method of calculating topological expansions of cost functions. A specific application to lubricated devices is illustrated.