A discontinuous Galerkin based enriched finite volume method

Abstract

In this report a new enriched finite volume method is introduced. This method will be particularly useful for fluid-structure interaction problems. In engineering many problems are in the domain of fluidstructure interaction. Examples of these are water in sea locks, blood flowing in vessels and sailing ships. Fluid problems are usually solved with the finite volume method (FVM), and structures in this fluid flow are handled by discretizing the fluid flow around this structure. The mesh therefore needs to be conforming around the structure, and remeshing is needed when the the structure is moving over time. Remeshing is a computationally expensive process, therefore in FEM a method called enriched FEM is developed, which removes the need for remeshing. Enriched FEM uses so-called enrichment functions; these enrichment functions decouple the structure from the mesh. This concept is implemented in the finite volume method to get an enriched FVM. In order to come up with this method; FVM and enriched FEM are compared to see if the concepts of enriched FEM can be implemented in FVM. The FE method that is used is the discontinuous Galerkin (DG) method, which is a method that uses components of FEM and FVM. It discretizes equations as in FEM, but the elements are connected using flux functions as in FVM. DG is used to develop a high-order FVM and to introduce enrichments in FVM. The result of this thesis is a method which can implement enrichment functions in FVM, such that remeshing is no longer needed. The method is compared with conforming FVM and enriched DG, and it uses less computational power than both methods. The method also recovers the optimal convergence rate for a non-conforming FVM discretization. The enrichment functions are decoupled from the FVM discretization, therefore the method can be added to existing FVM solvers without changing the non-enriched part. The method is tested on a problem where waves are generated in a box.