SUMMARYA methodology for solving three-dimensional crack problems with geometries that are independent of the mesh is described. The method is based on the extended ÿnite 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 branch functions include asymptotic near-tip ÿelds that improve the accuracy of the method. The crack geometry is described by two signed distance functions, which in turn can be deÿned by nodal values. Consequently, no explicit representation of the crack is needed. Examples for three-dimensional elastostatic problems are given and compared to analytic and benchmark solutions. The method is readily extendable to inelastic fracture problems.

SUMMARYWe present a level set method for treating the growth of non-planar three-dimensional cracks. The crack is deÿned by two almost-orthogonal level sets (signed distance functions). One of them describes the crack as a two-dimensional surface in a three-dimensional space, and the second is used to describe the one-dimensional crack front, which is the intersection of the two level sets. A Hamilton-Jacobi equation is used to update the level sets. A velocity extension is developed that preserves the old crack surface and can accurately generate the growing surface. The technique is coupled with the extended ÿnite element method which approximates the displacement ÿeld with a discontinuous partition of unity. This displacement ÿeld is constructed directly in terms of the level sets, so the discretization by ÿnite elements requires no explicit representation of the crack surface. Numerical experiments show the robustness of the method, both in accuracy and in treating cracks with signiÿcant changes in topology.

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