Generalized finite element enrichment functions for discontinuous gradient fields

A general GFEM/XFEM formulation is presented to solve two-dimensional problems characterized by C-0 continuity with gradient jumps along discrete lines, such as those found in the thermal and structural analysis of heterogeneous materials or in line load problems in homogeneous media. The new enrichment functions presented in this paper allow solving problems with multiple intersecting discontinuity lines, such as those found at triple junctions in polycrystalline materials and in actively cooled microvascular materials with complex embedded networks. We show how the introduction of enrichment functions yields accurate finite element solutions with meshes that do not conform to the geometry of the discontinuity lines. The use of the proposed enrichments in both linear and quadratic approximations is investigated, as well as their combination with interface enrichment functions available in the literature. Through a detailed convergence study, we demonstrate that quadratic approximations do not require any correction to the method to recover optimal convergence rates and that they perform better than linear approximations for the same number of degrees of freedom in the solution of these types of problems. In the linear case, the effectiveness of correction functions proposed in the literature is also investigated. Copyright (C) 2009 John Wiley & Sons, Ltd.