Enriched Reproducing Kernel Particle Approximation For Simulating Prob- lems Involving Moving Interfaces: Application To Solidification Problems

Finite element methods are commonly used to model melting problems where the problem domain is defined using a mesh support. When the domain evolves the mesh becomes very distorted, and remeshing as well as field projection between successive meshes are then required [5]. Moreover, to represent the thermal transmission conditions through the interface (melting or solidification fronts) the interface must be represent by facets (in 3D) or segments (2D) of the mesh skeleton, which involves continuous remeshing in the context of the FE approximations. Recently, several authors proposed to use meshless methods in order to get rid of the mesh-related problems. In fact, the term meshless methods covers several different methods, and we classify these methods by their ability to deal with the solidification problem according to the three following criteria: The first criterion is the way how the local discontinuous approximation is set up. In [2], Ji uses the Moving Least Squares (MLS) method associated with a Partition of Unity (PUM) enrichment, and in [8], Yvonnet uses the constrained natural method (CNEM). In a prior work, we proposed to use the enriched reproducing kernel particle approximation (ERKPA)[4, 3]. The second criterion is the way in which the problem equations are discretized. There are two possible ways. First, the weak formulation of the problem is used as in the EFG [1] or XFEM method. But these two cases require the use of an integration mesh and thus reduce the meshless character of the methods. Second, the strong formulation of the problem is used to discretize the problem as in the point collocation methods. In this paper, we use this last approach. The third criterion is the way how the solidification front is defined. The boundary can either be defined by using a mesh as in the CNEM method proposed by Yvonnet [8], or the front is defined totally independent from the nodes that define the approximation. For this latter case, Ventura [7] or Ji [2] proposed to use Level Set Methods [6].