A spatially adaptive phase-field model of fracture

Phase-field models of fracture introduce smeared cracks of width commensurate with a regularisation length parameter $\epsilon$ and obeying a minimum energy principle. Mesh adaptivity naturally suggests itself as a means of supplying spatial resolution were needed while simultaneously keeping the computational size of the model as small as possible. Here, a variational-based spatial adaptivity is proposed for a phase-field model of fracture. The conventional phase-field model is generalised by allowing a spatial variation of the regularisation length $\epsilon$ in the energy functional. The optimal spatial variation of the regularisation length then follows by energy minimisation in the same manner as the displacement and phase fields. The extended phase-field model is utilised as a basis for an adaptive mesh refinement strategy, whereby the mesh size is required to resolve the optimal length parameter locally. The resulting solution procedure is implemented in the framework of the finite element library FEniCS. Selected numerical tests suggest that the spatially adaptive phase-field model exhibits the same convergence rate as the conventional phase-field model, albeit with a vastly superior constant, which results in considerable computational savings.