Tamtam Proceedings - lamsin

Shape optimization 4051. IntroductionTopological optimization is concerned with the variation of a cost function with respectto a topology modification of a domain. The most simple way of modifying thetopology consists in creating a small hole in the domain. Usually, the cost function involvesthe solution of a p.d.e. defined on this domain. In the case of structural shapeoptimization, creating a hole means simply removing some material. In the case of fluiddynamics where the domain represents the fluid, creating a hole means inserting a smallobstacle. The situation is similar in electromagnetism. The topological sensitivity toolswhich have been developed by several authors [4, 7, 8] allow to find the place where creatinga small hole will bring the best improvement of the cost function. These tools arebased on a gradient like expression of the formj(Ω ε ) = j(Ω) + f(ε)g(x 0 ) + o(f(ε)), (1)where ρ(ε) is a scalar positive function going to zero with ε, Ω is an open and boundedsubset Ω of IR d , d = 2, 3, and, for ε > 0 and x 0 ∈ Ω, Ω ε = Ω\(x 0 + εω) is the subsetobtained by removing the subset x 0 + εω from Ω, where ω ⊂ IR d is a fixed open andbounded subset containing the origin. This expansion is called the topological asymptoticand g the topological gradient. The function g is very easy to compute. Obviously, ifwe want to minimize j, the "best" location where to create an infinitesimal hole is therewhere g(x) is the most negative.1ΓoutΓ inΓ2outΓ3outΓmoutFigure 1. The cavity Ω.In this work, we consider the Stokes equations describing the circulation of an incompressiblefluid flow in a cavity Ω having one inlet Γ in and some outlets Γ i out, i = 1, m.(see Figure 1). Our purpose is to determine the optimal shape of the tubes that connectthe inlet to the outlets of the cavity maximizing the outflow rate. It consists in insertingsome small obstacles in the cavity in order to maximize the outflow rate at Γ i out, i = 1, m.TAMTAM –Tunis– 2005