Optimization-based finite element methods for evolving interfaces

Abstract

This thesis is concerned with the development of new approaches to redistancing and conservation of mass in finite element methods for the level set transport equation. The first proposed method is a PDE- and optimization-based redistancing scheme. In contrast to many other PDE-based redistancing techniques, the variational formulation derived from the minimization problem is elliptic and can be solved efficiently using a simple fixed-point iteration method. Artificial displacements are effectively prevented by introducing a penalty term. The objective functional can easily be extended so as to satisfy further geometric properties. The second redistancing method is based on an optimal control problem. The objective functional is defined in terms of a suitable potential function and aims at minimizing the residual of the Eikonal equation under the constraint of an augmented level set equation. As an inherent property of this approach, the interface cannot be displaced on a continuous level and numerical instabilities are prevented. The third numerical method under investigation is an optimal control approach designed to enforce conservation of mass. A numerical solution to the level set equation is corrected so as to satisfy a conservation law for the corresponding Heaviside function. Two different control approaches are investigated. The potential of the proposed methods is illustrated by a wide range of numerical examples and by numerical studies for the well-known rising bubble benchmark.