- journal article metadata
European Mathematical Society Publishing House
2016-09-19 17:05:04
Interfaces and Free Boundaries
Interfaces Free Bound.
IFB
1463-9963
1463-9971
Partial differential equations
Numerical analysis
Fluid mechanics
Biology and other natural sciences
10.4171/IFB
http://www.ems-ph.org/doi/10.4171/IFB
subscribers, moving wall 5 years
European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society
18
2016
2
Shape optimization for surface functionals in Navier-Stokes flow using a phase field approach
Harald
Garcke
Universität Regensburg, REGENSBURG, GERMANY
Claudia
Hecht
Universität Regensburg, REGENSBURG, GERMANY
Michael
Hinze
Universität Hamburg, HAMBURG, GERMANY
Christian
Kahle
Universität Hamburg, HAMBURG, GERMANY
Kei Fong
Lam
Universität Regensburg, REGENSBURG, GERMANY
Shape optimization, phase-field method, lift, drag, Navier–Stokes equations
We consider shape and topology optimization of an object in fluid flow governed by the Navier–Stokes equations. Shapes are modelled with the help of a phase field approach and the solid body is relaxed to be a porous medium. The phase field method uses a Ginzburg–Landau functional in order to approximate a perimeter penalization. We focus on surface functionals and carefully introduce a new modelling variant, show existence of minimizers and derive first order necessary conditions. These conditions are related to classical shape derivatives by identifying the sharp interface limit with the help of formally matched asymptotic expansions. Finally, we present numerical computations based on a Cahn–Hilliard type gradient descent which demonstrate that the method can be used to solve shape optimization problems for fluids with the help of the new approach.
Calculus of variations and optimal control; optimization
Partial differential equations
219
261
10.4171/IFB/363
http://www.ems-ph.org/doi/10.4171/IFB/363