Shape optimization of a floating body

  • Gestaltsoptimierung eines schwimmenden Körpers

Franken, Norbert; Wagner, Alfred (Thesis advisor)

Aachen : Publikationsserver der RWTH Aachen University (2010)
Dissertation / PhD Thesis

Aachen, Techn. Hochsch., Diss., 2010

Abstract

In the present thesis we consider a variational formulation of a floating body in a perfect fluid. In particular, we study the case of a stationary two-dimensional potential flow and thus we are able to replace the velocity potential by its harmonic conjugate which is called the stream function. Compared to the velocity potential the stream function satisfies Dirichlet data instead of Neumann data on the boundary and we can characterize the liquid set as the positivity set of the stream function. The resulting energy functional depends on both stream function and floating body. We realize these two minimizations by working in two separated steps. When minimizing the stream function we work with the constraint that the volume of the fluid has to be constant. However, in order to use non-volume preserving perturbations we add a penalty term and disregard the volume condition. The choice of the penalty term yields an approximation of the original functional. Using direct methods we prove the existence of a minimizer of the penalized problem. Moreover, we show that the minimizer is bounded and subharmonic. By a construction of two appropriate comparison functions we get equivalence of the original and the penalized problem provided we have chosen adequate parameters. Furthermore, we use a technique of Morrey in order to show Hölder continuity and a method of Alt and Caffarelli in order to get Lipschitz continuity of the stream function, which is the maximal regularity which can be proved. A nondegeneracy property of the stream function leads to density estimates on the free surface of the fluid which imply that the free boundary has locally finite perimeter. We use the concept of blow-up limits to deduce gradient estimates on the surface of the fluid which finally lead to the statement that the reduced boundary is locally a C^{1,beta} surface. A further result which is only valid in a two-dimensional setting extends the regularity to the whole boundary. We look for the optimal floating body in the family of all compact sets with prescribed volume and a priori bounded number of connected components. In addition, we postulate boundedness of the density perimeter of the boundary in order to avoid oscillations that may eventually appear. Again we use direct methods and work with two different notions of domain convergence, namely Hausdorff convergence and gamma-convergence. In both cases we are able to show the existence of an optimal floating body.

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