The finite element method is now well established as a robust and reliable numerical technique in many areas of solid mechanics. There are however problems where the use of the finite element method is cumbersome like the modeling of moving discontinuities due to the need to update the mesh to match the geometry of the discontinuity. Even for stationary cracks, in three-dimensional solids, contsructing a mesh that matches the geometry of the crack is not trivial for non-symmetric loading cases. Recently, a new technique for modeling cracks in the finite element framework has been introduced. A standard displacement-based approximation is enriched near a crack by incorporating both a discontinuous field and the near crack front asymptotic fields through a partition of unity method. A methodology that constructs the enriched approximation from the interaction of the crack geometry with the mesh is developed. This technique allows the entire crack to be represented independently of the mesh, and so remeshing is not necessary to model crack growth. Numerical experiments are provided to demonstrate the utility and robustness of the proposed technique. Two-dimensional crack growth analysis are shown and stress intensity factors for planar three-dimensional crack obtained with the X-FEM are compared to available reference solutions from the literature.