Moving Domain by Galerkin-Level Set Strategy: Application to Shape Geodesics

In this paper, we use the concept of connecting tubes introduced in [7][5] and we consider the geodesic tube construction between two sets according to the tube shape metric. The first section is devoted to the tube analysis and formulation associated to the Galerkin-Level Set strategy. This new variational formulation consists in parameterizing the level set function in a finite vector space. Consequently, the aim of a Galerkin-Level Set method is more focused on topology that on high accuracy for the boundary approximation. However, the main advantage of this method, over the traditional level set formulation, concerns the standard partial differential equation (PDE) evolution for the level set function that, in the Galerkin-Level Set method turns into a system of ordinary differential equations and we avoid any “usual” instability. The second section concerns a shape identification problem associated to an Hilbert space metric using the Galerkin-Level Set method. In the last section, the geodesic tube construction is made by a optimization process based on a shape gradient calculus. Finally, a geodesic tube construction between two sets is validated by numerical experiments in 3D.