Adaptive fourth-order phase field method for rock fractures using novel refinement criteria and improved data transfer operators

High computing cost restricts the application of phase field models in geotechnical engineering (e.g., in blasting, oil and gas exploration, and rock landslides). To improve computational efficiency, this paper proposes an adaptive isogeometric method of the phase-field model for simulating rock fracture using a novel refinement criterion and an improved data transfer operator (HBFT). The proposed method is shown to decrease the calculation time and storage requirements by over 90% compared to the uniform refinement in most cases, and the computing time of incorporating non-equal order cells is 35.23% less than that of the equal order case. Notably: (1) the proposed refinement criterion is simple and efficient, and relies only on the 1D knot vector of the IGA to guarantee the hierarchical difference of adjacent cells.(2) the proposed HBFT only transfers the history variables in the local region to be refined, while keeping the variables in other regions unchanged; additionally, compared with the global and cell-by-cell versions in the traditional BFT, the proposed HBFT not only has the potential to avoid solving large-scale linear equations of the global version, but also alleviates, to a certain extent, the requirement of the cell-by-cell version for the full integration cell.