On a Space-Time Extended Finite Element Method for the Solution of a Class of Two-Phase Mass Transport Problems
Lehrenfeld, Christoph; Reusken, Arnold (Thesis advisor); Behr, Marek (Thesis advisor); Schöberl, Joachim (Thesis advisor)
Aachen / Publikationsserver der RWTH Aachen University (2015) [Dissertation / PhD Thesis]
Page(s): IX, 219 S. : Ill., graph. Darst.
In the present thesis a new numerical method for the simulation of mass transport in an incompressible immiscible two-phase flow system is presented. The mathematical model consists of convection diffusion equations on moving domains which are coupled through interface conditions. One of those conditions, the Henry interface condition, prescribes a jump discontinuity of the solution across the moving interface.For the description of the interface position and its evolution we consider interface capturing methods, for instance the level set method. In those methods the mesh is not aligned to the evolving interface such that the interface intersects mesh elements. Hence, the moving discontinuity is located within individual elements which makes the numerical treatment challenging. The discretization presented in this thesis is based on essentially three core components. The first component is an enrichment with an extended finite element (XFEM) space which provides the possibility to approximate discontinuous quantities accurately without the need for aligned meshes. This enrichment, however, does not respect the Henry interface condition.The second component cures this issue by imposing the interface condition in a weak sense into the discrete variational formulation of the finite element method. To this end a variant of the Nitsche technique is applied. For a stationary interface the combination of both techniques offers a good way to provide a reliable method for the simulation of mass transport in two-phase flows. However, the most difficult aspect of the problem is the fact that the interface is typically not stationary, but moving in time. The numerical treatment of the moving discontinuity requires special care. For this purpose a space-time variational formulation, the third core component of this thesis, is introduced and combined with the first two components: the XFEM enrichment and the Nitsche technique. In this thesis we present the components and the resulting methods one after another, for stationary and non-stationary interfaces. We analyze the methods with respect to accuracy and stability and discuss important properties. For the case of a stationary interface the combination of an XFEM enrichment and the Nitsche technique, the Nitsche-XFEM method, has been introduced by other authors. Their method, however, lacks stability in case of dominating convection. We combine the Nitsche-XFEM method with the Streamline Diffusion technique to provide a stable method also for convection dominated problems. We further discuss the conditioning of the linear systems arising from Nitsche-XFEM discretizations which can be extremely ill-conditioned.For the case of a moving interface we propose a space-time Galerkin formulation with trial and test functions which are discontinuous in time and combine this approach with an XFEM enrichment and a Nitsche technique resulting in the Space-Time Nitsche-XFEM method. This method is new. We present an error analysis and discuss implementation aspects like the numerical integration on arising space-time geometries.The aforementioned methods have been implemented in the software packages DROPS for the spatially three-dimensional case. The correctness of the implementation and the accuracy of the method is analyzed for test cases. Finally, we consider the coupled simulation of mass transport and fluid dynamics for realistic scenarios.