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.

DOI: 10.1051/cocv/2015006

Keywords: Shape and topology optimization, phase field method, diffuse interfaces, stationary Navier-Stokes flow, fictitious domain

^{1}; Hecht, Claudia

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@article{COCV_2016__22_2_309_0, author = {Garcke, Harald and Hecht, Claudia}, title = {Applying a phase field approach for shape optimization of a stationary {Navier-Stokes} flow}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {309--337}, publisher = {EDP-Sciences}, volume = {22}, number = {2}, year = {2016}, doi = {10.1051/cocv/2015006}, mrnumber = {3491772}, zbl = {1342.35218}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/cocv/2015006/} }

TY - JOUR AU - Garcke, Harald AU - Hecht, Claudia TI - Applying a phase field approach for shape optimization of a stationary Navier-Stokes flow JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2016 SP - 309 EP - 337 VL - 22 IS - 2 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2015006/ DO - 10.1051/cocv/2015006 LA - en ID - COCV_2016__22_2_309_0 ER -

%0 Journal Article %A Garcke, Harald %A Hecht, Claudia %T Applying a phase field approach for shape optimization of a stationary Navier-Stokes flow %J ESAIM: Control, Optimisation and Calculus of Variations %D 2016 %P 309-337 %V 22 %N 2 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2015006/ %R 10.1051/cocv/2015006 %G en %F COCV_2016__22_2_309_0

Garcke, Harald; Hecht, Claudia. Applying a phase field approach for shape optimization of a stationary Navier-Stokes flow. ESAIM: Control, Optimisation and Calculus of Variations, Volume 22 (2016) no. 2, pp. 309-337. doi : 10.1051/cocv/2015006. http://archive.numdam.org/articles/10.1051/cocv/2015006/

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