Improving the conditioning of XFEM/GFEM for fracture mechanics problems through enrichment quasi-orthogonalization

Partition of unity enrichment is known to significantly enhance the accuracy of the finite element method by allowing the incorporation of known characteristics of the solution in the approximation space. However, in several cases it can further cause conditioning problems for which a number of remedies have been proposed in the framework of the extended/generalized finite element method (XFEM/GFEM). Those solutions often involve significant modifications to the initial method and result in increased implementation complexity. In the present work, a simple procedure for the local near-orthogonalization of enrichment functions is introduced, which significantly improves the conditioning of the resulting system matrices, while requiring only minor modifications to the initial method. Although application to different types of enrichment functions is possible, the resulting scheme is specialized for the singular enrichment functions used in linear elastic fracture mechanics and tested through benchmark problems.