A novel approach to compute the spatial gradients of enriching functions in the X-FEM with a hybrid representation of cracks

Abstract

The eXtended Finite Element Method (X-FEM) is a versatile technique to model discontinuities by enriching the trial functions with a prior solution. In the X-FEM, a crack can be explicitly represented by a set of triangles or implicit signed distances, i.e., level set functions, of the points of interest from the crack surface and the crack front. In the explicit representations, it is crucial to accurately evaluate surface normal, conormal, and tangent vectors along crack fronts for computations of the gradients of the enriching functions. The solution is very sensitive to these directional vectors, especially for non-planar 3D cracks. We here propose a novel approach to compute these gradients without evaluating the directional vectors. Level set functions are first set up in a hexahedral grid independent of the background mesh in the X-FEM. We can thus implement the Weight Essentially Non-Oscillatory (WENO) scheme to compute the gradients of level sets. The gradients of the enriching functions at any integration point can therefore be computed by interpolations and chain rules. We compare the implementation procedures of the explicit representation and the proposed hybrid representation in detail. A three-dimensional lens crack problem is studied to demonstrate the accuracy of the proposed method, especially for coarse meshes.