The use of Powell-Sabin B-Splines in a higher-order phase-field model for crack kinking

Phase-field models for brittle fracture in anisotropic materials result in a fourth-order partial differential equation for the damage evolution. This necessitates a 𝒞^1 continuity of the basis functions. Here, Powell-Sabin B-splines, which are based on triangles, are used for the approximation of the field variables as well as for the the description of the geometry. The use of triangles makes adaptive mesh refinement and discrete crack insertion straightforward. Bézier extraction is used to cast the B-splines in a standard finite element format. A procedure to impose Dirichlet boundary condition is provided for these elements. The versatility and accuracy of the approach are assessed in two case studies, featuring crack kinking and zig-zag crack propagation. It is also shown that the adaptive refinement well captures the evolution of the phase field.