Abstract

One type of singularity that may appear in a three-dimensional fracture problem under elastic behavior is the free corner singularity, which occurs at the intersection of the crack with a free boundary and whose effect is usually ignored. The extended finite element method (XFEM) is a technique that allows the efficient numerical modeling of fracture problems, by using enrichment functions within a finite element model that incorporate the geometry and effect of the crack. However, when applied to the study of a problem with corner singularity, the singular behavior is not properly captured. The reason is that the usual enrichment in the XFEM only describes the typical crack front singularity. Hence, in order to include the effect of the free border singularity, the enrichment has to be modified. In this work, we present a new set of functions which relies on spherical harmonics and is able to capture the behavior of the free border singularity.

Abstract

The simulation of immiscible two-phase flows on Eulerian meshes requires the use of special techniques to guarantee a sharp definition of the evolving fluid interface. This work describes the combination of two distinct technologies with the goal of improving the accuracy of the target simulations. First of all, a spatial enrichment is employed to improve the approximation properties of the Eulerian mesh. This is done by injecting into the solution space new features to make it able to correctly resolve the solution in the vicinity of the moving interface. Then, the Lagrangian Particle Level Set (PLS) method is employed to keep trace of the evolving solution and to improve the mass conservation properties of the resulting method. While the local enrichment can be understood in the general context of the XFEM, we employ an element-local variant, which allows preserving the matrix graph, and hence highly improving the computational efficiency.