# Three Dimensional CS-FEM Phase-Field Modeling Technique for Brittle Fracture in Elastic Solids
## 4 Discussion  
The primary objective of this study was to predict realistic crack paths in three-dimensional solids. The CS-FEM-H8 element is implemented in the commercial platform of ABAQUS. The element was already proven to have a very good accuracy in displacement norm [38]; here, we combined it with the phase-field model for diffuse fracture to simulate crack propagation in 3D linear elastic brittle solids. ABAQUS user elements (UEL) has its inherent limitations in not letting the user access data  
Figure 14: Crack propagation in a thick-walled cylinder: (**a**) isometric view of the cylinder; and (**b**) relative position of the two Cracks. Figure 15: Crack propagation in a thick-walled cylinder: (**a**) crack initiation; (**b**) pattern of the two cracks propagating; and (**c**) final crack merging. from surrounding elements, thus we developed a much simpler technique to calculate stress/strain for smoothing domains instead of the traditional weighted average approach. We sacrificed the stress convergence rate due to this new formulation, however still obtained accurate results that predict complex crack topologies such as crack branching and curved crack paths. Several drawbacks of the method were observed during the simulations: the phase-field variable propagation is very much mesh dependent, thus one needs to have a very fine mesh (approximate characteristic element size of \((1/2)\) the length scale parameter, or less [24; 25; 28]) near the expected crack propagation zone to efficiently resolve the length-scale parameter. This is not an issue for geometries where experimentally obtained crack paths are already known, however, for complex geometries and unknown crack paths, we needed to have a very fine mesh throughout the domain to facilitate independent crack propagation which increases the computational cost. Once the length scale parameter has been successfully resolved, further reduction of the mesh size does not alter the crack path. We also observed that the selection of displacement increment step size affects the resultant crack path. The final fracture pattern is not dependent on the Griffith's energy release rate parameter, however the ultimate strength of the sample decreases with decreasing \(G_{c}\). Implementation in ABAQUS, as already discussed, has its own drawbacks in not letting the user implement more accurate S-FEM models such as ES-FEM/\(\alpha\)-FEM [18]. This formulation was developed based on the H8 element with eight smoothing cells, which was not generated automatically for complex geometries. Analysts probably need to resort to the use of tetrahedral (T4) elements in cases of complex topologies. This in turn affects the accuracy because the formulations of the T4 elements are the same for both CS-FEM and FEM [1]. However, this opens up an avenue for future research in this domain, where in-house codes can be used to solve the same problem using a much more accurate ES-FEM/FS-FEM and a base mesh of T3/T4 elements, which are very easily generated for complex geometries. Moreover, adaptive meshing schemes designed based on values of the phase-field parameter can be utilized to reduce the computational cost of having very fine mesh. In 2D cases, CS-FEM is slightly more efficient than FEM [39]. However, in this 3D setting, the computational cost is similar (quantitative difference of around 1%) for both FEM and CS-FEM formulations. This is due to the assembly process of the stiffness matrix for each smoothing domain which involves integration over each surface of the cell. However, the better accuracy in the displacement norm and the comparative \"soft\" property in stiffness matrix with respect to FEM gives CS-FEM an advantage. Thus, it can be concluded that this method is another efficient and novel computational technique to model crack propagation in 3D structures. Formal analysis, S.B. ; Investigation, S.B. ; Methodology, S.B. ; Supervision, G.-R.L. ; Validation, S.B. ; Writing--original draft, S.B. ; and Writing--review and editing, S.B. and G.-R.L.