Interfaces and Free Boundaries


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Volume 18, Issue 2, 2016, pp. 219–261
DOI: 10.4171/IFB/363

Published online: 2016-09-13

Shape optimization for surface functionals in Navier-Stokes flow using a phase field approach

Harald Garcke[1], Claudia Hecht[2], Michael Hinze, Christian Kahle and Kei Fong Lam[3]

(1) Universität Regensburg, Germany
(2) Universität Regensburg, Germany
(3) Universität Regensburg, Germany

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.

Keywords: Shape optimization, phase-field method, lift, drag, Navier–Stokes equations

Garcke Harald, Hecht Claudia, Hinze Michael, Kahle Christian, Lam Kei Fong: Shape optimization for surface functionals in Navier-Stokes flow using a phase field approach. Interfaces Free Bound. 18 (2016), 219-261. doi: 10.4171/IFB/363