A well-conditioned and optimally convergent XFEM for 3D linear elastic fracture

A variation of the extended finite element method for three-dimensional fracture mechanics is proposed. It utilizes a novel form of enrichment and point-wise and integral matching of displacements of the standard and enriched elements in order to achieve higher accuracy, optimal convergence rates, and improved conditioning for two-dimensional and three-dimensional crack problems. A bespoke benchmark problem is introduced to determine the method's accuracy in the general three-dimensional case where it is demonstrated that the proposed approach improves the accuracy and reduces the number of iterations required for the iterative solution of the resulting system of equations by 40% for moderately refined meshes and topological enrichment. Moreover, when a fixed enrichment volume is used, the number of iterations required grows at a rate which is reduced by a factor of 2 compared with standard extended finite element method, diminishing the number of iterations by almost one order of magnitude.