A consistent phase-field-regularised partition of unity method for fracture analysis

Recent advancements in phase-field models have significantly reshaped the landscape of fracture mechanics, which was dominated by the partition of unity method in the early 21st century. In this study, we aim to leverage the advantages of the two approaches by adopting a novel phase-field-regularised partition of unity method to improve computational efficiency, robustness and physical consistency. Specifically, we establish a connection between early phase-field models and the partition of unity method for cohesive fracture. To this end, we replace the standard discontinuous Heaviside enrichment in the partition of unity method with a regularised and continuous Heaviside function, leveraging the phase-field approximation of the Dirac-δ function. The proposed formulation effectively resolves ill-conditioning issues in the traditional partition of unity method while retaining the key advantages of discrete fracture representations, offering a distinct contrast to traditional phase-field approaches for smeared crack models. These advantages include eliminating the need for extremely fine meshes and providing an unambiguous and physically consistent representation of the displacement jump across a crack. Furthermore, by integrating Non-Uniform Rational B-Splines (NURBS) for spatial discretisation, the approach enhances solution accuracy compared to standard finite element formulations. Compatibility enforcement is also modified to accommodate the crack diffused by the phase-field approximation. Through numerical examples, including stationary and propagating cracks, mesh refinement studies, and sensitivity analyses of the phase-field length scale, we establish an optimal prescription for the internal length scale based solely on the element size. The examples compare the results obtained via the presented formulations with exact solutions and other numerical techniques, demonstrating the accuracy, conditioning stability, and computational efficiency of the methodology. The proposed methodology thus presents a robust alternative to conventional fracture models, combining key advantages offered by discrete and smeared approaches.