Topological asymptotic expansions for the generalized Poisson problem with small inclusions and applications in lubrication
Gustavo C. Buscaglia, Ionel Sorin Ciuperca, Mohammed Jai · Inverse Problems · 2007
The asymptotic expansion of the solution u to the equation ∇ ⋅ (α∇ u ) = −∇ ⋅ β + γ in , with respect to the size of an inclusion (at a point x 0 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 − u = n z , it is shown that the sequence z converges weakly, for all p 0, to a function z in L p (Ω)∩ H 1 (Ω ∖ B ρ ( x 0 )), where B ρ ( x 0 ) is the ball of radius ρ around x 0 . This allows for the calculation of asymptotic expansions of cost functions of the form J ( ) = ∫ Ω F ( u ) d x , 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.