Applying a phase field approach for shape optimization of a stationary Navier-Stokes flow
Revised: 2 February 2015
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.
Mathematics Subject Classification: 35R35 / 49Q10 / 49Q12 / 49Q20 / 76D05
Key words: Shape and topology optimization / phase field method / diffuse interfaces / stationary Navier-Stokes flow / fictitious domain
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