Tamtam Proceedings - lamsin

Shape optimization 4072.1. Variation of the cost function when inserting a small obstacleThe topological sensitivity results for the Stokes equations, when inserting a smallobstacle inside the domain are the following. We have to distinguish the cases d = 2 andd = 3. This is due to the fact that the fundamental solutions to the Stokes equations inIR 2 and IR 3 have an essentially different asymptotic behavior at infinity. It is proved in[5, 6] that f(ε) = ε if d = 3 and f(ε) = −1/ log(ε) if d = 2.2.1.1. The three dimensional caseTheorem 2.1 If Hypothesis 2.1 holds, then function j has the following asymptotic expansion[( ∫)]j(ε) = j(0) + ε − T (y) ds(y) .v 0 (x 0 ) + δJ + o(ε) (4)∂ωwhere T ∈ H −1/2 (∂ω) is solution to the boundary integral equation∫E(x − y) T (y) dγ(y) = −u 0 (x 0 ), ∀x ∈ ∂ω.∂ωThe function v 0 is the solution to the adjoint problem associated to (3) and the function(E, P ) is the fundamental solution to the Stokes equations in 3D:E(y) = 1 ( )I + e r e T r , P (y) = y8πνr4πr 3 , with r = ||y|| and e r = y/r.In the particular case where ω = B(0, 1), the density T is given explicitly T (y) =3ν2 u 0(x 0 ), ∀y ∈ ∂ω.Corollary 2.1 Let x 0 ∈ Ω. Under the hypotheses of theorem 2.1, we have[]j(ε) = j(0) + ε 6πν u 0 (x 0 ).v 0 (x 0 ) + δJ + o(ε). (5)2.1.2. The two dimensional caseIn this case the fundamental solution (E, P ) of the Stokes equations is given byE(y) = 1 ()− (log(r)I + e r e T r , P (y) = y4πν2πr 2 .Theorem 2.2 Under the same hypotheses of theorem 2.1, the function j has the followingasymptotic expansion[]j(ε) = j(0) + −1/ log(ε) 4πν u 0 (x 0 ).v 0 (x 0 ) + δJ + o(−1/ log(ε)). (6)TAMTAM –Tunis– 2005