Wave Function Spartan 10 Crack 12 [TOP]
For elements completely cut by the crack, the jump function is often chosen to cause a discontinuous displacement field with where denotes the normal vector to the crack surface and is a level set function; see Figure 12. The zero-isobar describes the position of the crack surface. Often a shifting is employed, see the second term on the RHS of (14) and Figure 11, and ensures that the enriched shape function vanishes in the neighboring element. The shifting also maintains the interpolatory character of the XFEM approximation or in other words: the nodal parameters remain the physical displacements.
wave function spartan 10 crack 12
Although the XFEM approximation is capable of representing crack geometries that are independent of element boundaries, it relies on the interaction between the mesh and the crack geometry to determine the sets of enriched nodes . This leads to particular crack configurations that cannot be accurately be represented by (13). Such cases are shown in Figure 16. As the crack size approaches the local nodal spacing, the set of nodes for the Heaviside or step enrichment is empty, Figure 16(b). Moreover, node 1 for the cracking case in Figure 16(a) or nodes 1 to 4 for the cracking case in Figure 16(b), respectively, contain two branch enrichments. Thus, the standard approximation gets difficulties with this crack configuration since the discontinuous function extends too far. This problem always arises when one or more nodal supports contain the entire crack geometry. Usually, this kind of problem arises whenever a crack nucleates. Similar difficulties occur for approximations without any crack tip enrichment; see Figure 16(d). In order to close the crack within a single element, the set is empty as well. One solution is to refine the mesh locally such that the characteristic element size falls below that of the crack. An admissible crack configuration is shown in Figure 17.
The visibility method is the first approach that introduces a discrete crack into the meshfree discretization. In the visibility method, the crack boundary is considered to be opaque. Thus, the displacement discontinuity is modeled by excluding the particles on the opposite side of the crack in the approximation of the displacement field, see Figure 19: This is identical to setting the shape function across the crack to zero as shown in Figure 19. The jump in the displacement is then computed by Difficulties arise for particles close to the crack tip since undesired interior discontinuities occur (Figure 20) due to the abrupt cut of the shape function, see Figure 19. Nevertheless, Krysl and Belytschko  showed convergence for the visibility method. For linear complete EFG shape functions, they even showed that the convergence rate is not affected by the discontinuity. The length and size of the (undesired) discontinuities depend on the nodal spacing near the crack boundary. If the nodal spacing approaches zero, the length of the discontinuities tends to zero. With this argument and the theory of nonconforming finite elements, convergence of the discontinuous displacement field can be shown.
It should also be noted that the visibility criterion leads to discontinuities in shape functions near nonconvex boundaries such as kinks, crack edges, and holes, as shown in Figure 20 in two dimensions.
The diffraction method is an improvement of the visibility method. It eliminates the undesired interior discontinuities; see Figure 21 (see also Figure 19). The diffraction method is also suitable for non-convex crack boundaries. It is motivated by the way light diffracts around a sharp corner, but the equations used in constructing the domain of influence and the weight function bear almost no relationship to the equation of diffraction. The method is only applicable to radial basis kernel functions with a single parameter .
The idea of the diffraction method is not only to treat the crack as opaque but also to evaluate the length of the ray by a path which passes around the corner of the discontinuity. A typical weight function is shown in Figure 19. It should be noted that the shape function of the diffraction method is quite complex with several areas of rapidly varying derivatives that complicates quadrature of the discrete Galerkin form . Moreover, the extension of the diffraction method into three dimensions is complex. Nevertheless, several authors have presented implementations of the diffraction method in 3D [216, 217].
An additional requirement is usually imposed for particles close to the crack. Since the angle between the crack and the ray from the node to the crack tip is small, a sharp gradient in the weight function across the line ahead of the crack is introduced. In order to reduce this effect, Organ et al.  imposed that all nodes have a minimum distance from the crack surface, that is, the normal distance to the crack surface must be larger than with ; they suggested .
An extended element-free Galerkin (XEFG) method based on vector level sets was first proposed by Ventura et al.  in the context of LEFM. It was later extended to nonlinear materials and cohesive cracks by Rabczuk and Zi , Rabczuk et al. . In contrast to XFEM, due to the heavily overlapping shape functions, a crack tip enrichment has to be employed to ensure crack closure at the crack tip though alternative methods have been developed that avoid the use of tip enrichments [229, 230]. However, those approaches are cumbersome in three dimensions.
The approximation of the displacement field in the cracking particles method is given by where is the total set of nodes in the model and is the set of cracked nodes; is the step function. In general, different shape functions can be used for the continuous and discontinuous part of the displacement field.
Numerical experiments on the proposed formulation were carried out by Bourdin et al. , who pointed to the similarity of the new formulation to models of image segmentation obtained by minimization of the Mumford-Shah functional. This meant that the inapplicability of standard numerical methods, owing to the fact that the formulation allowed for jump sets of the displacement field (representing the cracks) whose locus was unknown a priori. Two methods of solution were demonstrated, based on the mathematical concept of -convergence. In one of these methods, a second field was included in the energy functional in addition to the primary displacement field. This auxiliary scalar field acted as a regularizer for the jump sets of the displacement field, and took on values of 0 on the crack and 1 away from it. The regularizing variable was referred to by later researchers as the phase field, and it allowed for the treatment of the global energy minimization as a standard variational problem for which classical FEM is up to the task, albeit with the restriction that the characteristic length of the mesh should tend to zero faster than the characteristic length of the regularization so as not to overestimate the surface energy of the crack. Due to the inclusion of a second field, a coupled system of equations must now be solved consisting of the original equilibrium/linear momentum equations and the evolution PDE of the phase field.
A phase field model for mode III dynamic fracture was devised by Karma et al. , which made use of a phase field evolution equation based on the standard two-minimum Ginzburg-Landau form, that is, where is a function obeying the constraints , , and . The KKL phase field model was subsequently utilized by Hakim and Karma  to analyze the laws of quasistatic crack tip motion. Among other things, they showed that for kinked cracks in anisotropic media, force-balance gives predictions that are significantly different from those obtained using the principle of maximum energy release rate, and that the predictions obtained from force-balance hold even when fracture is modeled as irreversible. It should be mentioned that up to this point, phase field implementations featured isotropic damage wherein fracture occurred both in tension and compression. This sometimes resulted in unphysical response, such as sample interpenetration in the crack branching simulation in Bourdin et al. .
Thermodynamically consistent phase field models of fracture were developed by Miehe et al. [266, 267], along with corresponding incremental variational principles and numerical implementation within a finite element framework. Their implementation of the phase field was also slightly different from that of previous authors whereas Bourdin et al.  and subsequent papers utilized the convention that the phase field took values of 0 at cracks and 1 at unbroken states (hence a pseudo material density function) and Miehe et al. reversed the convention by assigning to the phase field values of 1 and 0 for fully cracked and fully unbroken states respectively. As a consequence, the functional associated with the phase field may now be seen to represent the crack surface itself. They then derived the evolution equation for the phase field based on the assumption that the solution is negative exponential in nature. Furthermore, they extended the application of phase fields to fracture simulations involving viscous overforce response via a time-regularized three-field formulation, and showed that the coupled system (linear momentum + phase field evolution) could be solved either monolithically or via a robust staggered-update scheme [266, 267].
An important addition of Miehe et al. [266, 267] to the existing theory was the modelling of anisotropic degradation by using an additive decomposition of the stored energy of the undamaged solid, to come up with an anisotropic energy function of the form where the positive and negative parts of the energies are defined by and are ramp functions defined as . The energy decomposition allows for the case where fracture occurs in tension only as opposed to both tension and compression, and effectively fixes the sample inter-penetration issue encountered by Bourdin in crack branching simulations. Following the reversed convention on the phase field, the constraints on are , , and . To maintain numerical stability, a small positive constant is included in the formulation so that the material retains some residual stiffness even when it attains a fully damaged state.