Volume-Preserving Continuous Galerkin Level-Set Approach for Linear Finite Elements

Many physical phenomena can be described as the evolution of two phases coexisting within the same domain. Examples of such phenomena are the transport of gas and oil, solidification and phase transformations. Each of these phenomena require a description of the dynamics under which the phases change and a technique for tracking the interface between the relevant phases. Currently several techniques exist for describing the evolution of an interface between two phases. We can distinguish these techniques as either interface-tracking or interface-capturing. Interface-tracking techniques commonly describe the interface exactly, for example by representing the interface explicitly in a mesh [15], by assuming a parametric shape of the interface (See for example [19]) or by introducing markers indicating one of the phases [7] or the interface [4], and tracking explicitly the evolution of this interface. The class of interface-capturing techniques describe the interface implicitly by a function, such as the level-set method [12], the volume-of-fluid method [9] and the moment-of-fluid method [5], and track the evolution of this function explicitly. Recently, both methods have been combined to exploit the ease of capturing the location of the interface from the level-set method and the volume-preserving capacities of the volume-of-fluid method. So far, this coupling has only been performed on regular quadrilateral meshes adopting finite-volume discretisations [17] and on triangular meshes in the context of discontinuous-Galerkin finite-elements. In this article we develop a volume-preserving level-set method by coupling a Galerkin level-set formulation based on linear triangles with the volume-of-fluid method on star-shaped polygonal finite-volume meshes. This article will first introduce introduce the level-set method and the volume-of-fluid method and our choice of discretisation for each of the methods. Subsequently we will define the coupling between the two methods and the novel volume-correction algorithm which will ensure volume preservation during advection of the level-set and volume-of-fluid functions. Finally we will investigate the numerical and practical aspects of the volume-correction algorithm by several examples.