A corrected XFEM approximation without problems in blending elements

Abstract

SUMMARYThe extended finite element method (XFEM) enables local enrichments of approximation spaces. Standard finite elements are used in the major part of the domain and enriched elements are employed where special solution properties such as discontinuities and singularities shall be captured. In elements that blend the enriched areas with the rest of the domain problems arise in general. These blending elements often require a special treatment in order to avoid a decrease in the overall convergence rate. A modification of the XFEM approximation is proposed in this work. The enrichment functions are modified such that they are zero in the standard elements, unchanged in the elements with all their nodes being enriched, and varying continuously in the blending elements. All nodes in the blending elements are enriched. The modified enrichment function can be reproduced exactly everywhere in the domain and no problems arise in the blending elements. The corrected XFEM is applied to problems in linear elasticity and optimal convergence rates are achieved.

The extended/generalized finite element method: An overview of the method and its applications

“…Recently an effective and simple approach has been proposed by Fries [45] who used a linearly decreasing weight function for the enrichment in the blending elements. This approach allows one to obtain a conforming approximation and to eliminate partially enriched elements, so that the partition of unity property is everywhere satisfied.…”

A finite element approach to the simulation of hydraulic fractures with lag

“…This is particularly important for the crack tip singularity because the convergence of the solution with the mesh size could be rather slow otherwise C··..I hlf2 for the stresses). A useful alternative, which we have tested, is to include the crack tip singularities as enrichment shape functions, as in [34,35,27], Such an approach can recover an optimal order of convergence (1"' ..1 h for stresses) and enables the use of less refined meshes near the cracktip. Of course, it does not help with steep pressure gradients when L 1"' ..1 a.…”

Two‐Scale Simulation of a Disc With a Crack by X‐FEM and Cell Model

“…To overcome this deficiency, the nodes outside the radius of the blending elements are also enriched with the near tip solution, see Fig. 1 and the corrected X-FEM approach proposed in [4]. The four enrichment functions F k with k = 1, .…”