Applying a phase field approach for shape optimization of a stationary Navier-Stokes flow

Garcke, Harald; Hecht, Claudia

ESAIM: Control, Optimisation and Calculus of Variations, Tome 22 (2016) no. 2, pp. 309-337.

We apply a phase field approach for a general shape optimization problem of a stationary Navier-Stokes flow. To be precise we add a multiple of the Ginzburg–Landau energy as a regularization to the objective functional and relax the non-permeability of the medium outside the fluid region. The resulting diffuse interface problem can be shown to be well-posed and optimality conditions are derived. We state suitable assumptions on the problem in order to derive a sharp interface limit for the minimizers and the optimality conditions. Additionally, we can derive a necessary optimality system for the sharp interface problem by geometric variations without stating additional regularity assumptions on the minimizing set.

Classification : 35R35, 49Q10, 49Q12, 49Q20, 76D05
Mots clés : Shape and topology optimization, phase field method, diffuse interfaces, stationary Navier-Stokes flow, fictitious domain

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