Mutual Enhancement in Blending Partition of Unity and Phase-Field Methods

  • Fathi, Farshid (University of Sheffield)
  • de Borst, René (University of Sheffield)
  • Torelli, Giacomo (University of Sheffield)

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Once dominated by enriched discrete fracture models, the landscape of fracture mechanics has been significantly reshaped by Phase-Field models over the past decades. Although substantial efforts have been devoted to studying various aspects of the two approaches, comparatively little attention has been paid to establishing a connection between them – particularly in highlighting their similarities and exploring possible interactions that could help mitigate their respective shortcomings. Herein, we aim to demonstrate mutual benefits of leveraging Partition of Unity and Phase-Field methods within a unified framework – which, for instance, resolves the ill-conditioning issues commonly associated with traditional Partition of Unity method while retaining the key advantages of discrete fracture representations. At the same time, the framework eliminates the need for extremely fine meshes typically required in Phase-Field models and provides an unambiguous, physically consistent representation of the displacement jump across a crack, which is pivotal in cohesive fracture models. To this end, we replace the standard discontinuous Heaviside enrichment in the Partition of Unity method with a regularised, continuous Heaviside function constructed using the Phase-Field approximation of the Dirac-delta function. The proposed methodology provides a new perspective for computational fracture mechanics and offers the potential for a robust alternative to conventional fracture models by combining the key advantages of discrete and smeared approaches.