Adaptive phase field analysis with dual hierarchical meshes for brittle fracture

We present a novel dual-mesh based adaptive phase field method for solving fracture problems. As the name suggests, the proposed scheme is solved using two meshes with different characteristic element sizes (h); a coarser mesh for the elastic field and a finer mesh for the phase field. To facilitate the exchange of information between the meshes, an efficient data transfer algorithm is proposed. The efficiency of the formulation is enhanced by implementing an adaptive h-refinement scheme using polynomial splines over hierarchical T-meshes (PHT-splines). The inherent hierarchical nature of PHT-Splines and their localization property is exploited in the proposed data transfer algorithm as well as in implementing adaptivity. Independent refinement strategies have been proposed for both meshes. To illustrate the performance of the proposed approach, six numerical examples have been presented. For most examples, the critical load and computational efficiency (in terms of CPU time) have been reported for different combinations of element sizes of both meshes. The present work concludes that the proposed approach with h = 0.5l(0) for the elastic mesh and h = 0.25l(0) for the phase field mesh yields results as accurate as results obtained from the conventional phase field model with h = 0.25l(0), where l(0) denotes length scale parameter. Moreover, the elastic analysis being performed on a coarser mesh makes the proposed approach faster and cheaper than the conventional approach, indicating potential for future applications in other phase field problems.