Tamtam Proceedings - lamsin

406 Maatoug2. Topological optimization problemConsider a viscous incompressible fluid F, governed by Stokes equations, in steadyregime. The Eulerian velocity vector u and the pressure p of F fulfil the system⎧⎨⎩−ν∆u + ∇p = F in Ωdiv u = 0 in Ωu = u d on Γ,where Ω ⊂ IR d , d = 2, 3, is the domain occuped by the fluid, ν is the (constant) kinematicviscosity coefficient of F, F is a given body force per unit of mass and u d is prescribedvelocity on the boundary Γ = ∂Ω.We denote by Ω ε the perturbed domain, obtained from inserting a small obstacle ω εin Ω. We suppose that the obstacle has the form ω ε = x 0 + εω, where x 0 ∈ Ω, ε > 0 andω is a given fixed and bounded domain of IR d , containing the origin, whose boundary ∂ωis connected and piecewise of class C 1 .In Ω ε , the velocity u ε and the pressure p ε of F are solution to⎧⎪⎨⎪⎩−ν∆u ε + ∇p ε = F in Ω εdiv u ε = 0 in Ω εu ε = u d on Γu ε = 0 on ∂ω ε ,Note that for ε = 0, Ω 0 = Ω and (u 0 , p 0 ) is solution to⎧⎨ −ν∆u 0 + ∇p 0 = F in Ω,div u 0 = 0 in Ω,⎩u 0 = u d on Γ.Consider now a cost function j(ε) of the formj(ε) = J ε (u ε ),where J ε is defined on H 1 (Ω ε ) d for ε ≥ 0 and J 0 is differentiable with respect to u, itsderivative being denoted by DJ 0 (u).Suppose that J satisfies the following hypothesis.(2)(3)Hypothesis 2.1 There exist a function δJ defined on H 1 (Ω) d and a positive scalar functionf(ε) such that()J ε (v) − J 0 (u) = DJ 0 (u)(v − u) + f(ε)δJ(u) + o ‖v − u‖ 1,Ω+ f(ε) u, v ∈ H 1 (Ω) dlim f(ε) = 0.ε→0Our aim is to derive an asymptotic expansion of the cost function j with respect to ε.TAMTAM –Tunis– 2005