Higher order enriched FEM for cracked 3d solids

Abstract

In structural mechanics, problems involving features such as discontinuities and singularities on a finite element mesh can be effectively solved with the extended and generalized finite element methods (XFEM/GFEM), which employ a partition of unity (PU) enrichment to extend the finite element approximation space with functions tailored to describe the solution of the singularities. In these procedures, discontinuities can be represented implicitly by employing a set of signed distance functions, greatly facilitating the detection of their interfaces with the elements. With these methods, loss of optimal convergence due to the presence of singularities can be avoided with the use of appropriate enrichment schemes. Nevertheless, in several cases, linear enriched finite elements still underperform with respect to non-enriched higher order finite elements, despite benefitting from optimal convergence. In light of this fact, the use of higher order enriched finite elements becomes an attractive approach, since it can provide both better accuracy and higher order convergence rates. In the present contribution, we apply higher order finite elements with discontinuous and singular enrichment functions in the solution of cracked solid problems. Extending the enrichment approach to the three-dimensional, higher order case introduces challenges that can generally be grouped in the following two categories: problems related to the conditioning of the resulting system matrix, and reliably describing the higher order geometries of the element partitions necessary for numerical integration. We address the first category of issues by employing existing techniques from the literature; while for the second, the higher order representations of the interfaces arise directly from interpolating the implicit signed distances with the element shape functions. To ensure the topological validity of the cut partitions, a recursive binary tree algorithm is employed to describe the zero iso-surface with increasing refinement where the geometries involved present stronger curvatures. Finally, a set of numerical examples involving both planar cracks with curved crack fronts and non-planar cracks is presented to assess the performance of the proposed method.