Modeling Stationary and Evolving Discontinuities with Finite Elements

A methodology for treating non-planar three-dimensional cracks with geometries
that are independent of the mesh is summarized. The method is based on the extended finite element method, in which the crack discontinuity is introduced as a Heaviside step function via a partition of unity. In addition, branch functions are introduced for all elements containing the crack front. The crack geometry is described by two signed distance functions (level sets), which in turn can be defined by nodal values. Consequently, no explicit representation of the crack is needed. A Hamilton-Jacobi equation is used to update the level sets as the crack grows.
Numerical experiments show the robustness of the method in treating cracks with significant changes in topology. The method is readily extendable to inelastic fracture problems.