Surface Representation with Closest Point Projection in the X-FEM

Abstract

At present, the eXtended Finite Element Method (X-FEM) is a common generalization of the classical finite element method for solving problems of deformable solid mechanics in the presence of large cracks. The main advantage of this method is the possibility of using computational meshes that are not consistent with the crack geometry and the feasibility of accurately taking into account the singular asymptotics of the solution near the front of the crack. One of the key elements of this method is the way that the middle surface of the crack is represented in the algorithm. It is traditional to use an implicit surface representation based on the level set method. Such an approach is efficient and robust and allows the calculation in the case of propagating cracks. In the paper, we propose a version of the X-FEM approach which uses the closest point projection (CPP) method to represent the middle surface of the crack and provides, in our opinion, several advantages over the common X-FEM. The paper presents a short overview of the classical X-FEM technique. The proposed algorithm and its distinctions from the conventional one are described in detail and its advantages are formulated. We consider the problems of the numerical integration of functions defined on the surface described by the closest point projector method, the local recovery of the level set functions, and the computation of local frames at a point of the surface or its edge. The algorithmic details of the X-FEM enriched with a surface representation based on the projection of the closest point are described. In conclusion, we present some numerical results that demonstrate the algorithmic features of the method and performance capabilities of the proposed algorithm.