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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.
Garcke, Harald 1 ; Hecht, Claudia 1
@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://geodesic.mathdoc.fr/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://geodesic.mathdoc.fr/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://geodesic.mathdoc.fr/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, Tome 22 (2016) no. 2, pp. 309-337. doi: 10.1051/cocv/2015006
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