Shape optimization and subdivision surface based approach to solving 3D Bernoulli problems

Abstract

In the paper we consider a treatment of Bernoulli type shape optimization problems in three dimensions by the combination of the boundary element method and the hierarchical algorithm based on the subdivision surfaces. After proving the existence of the solution on the continuous level we discretize the free part of the surface by a hierarchy of control meshes allowing to separate the mesh necessary for the numerical analysis and the choice of design parameters. During the optimization procedure the mesh is updated starting from its coarse representation and refined by adding design variables on finer levels. This approach serves as a globalization strategy and prevents geometry oscillations without any need for remeshing. We present numerical experiments demonstrating the capabilities of the proposed algorithm.