A spatially adaptive phase-field model of fracture

TL;DR

This paper introduces a variational-based spatial adaptivity for phase-field fracture models by allowing the regularisation length to vary spatially, leading to improved computational efficiency without sacrificing convergence.

Contribution

It generalizes the phase-field model to include spatially varying regularisation length and develops an adaptive mesh refinement strategy based on energy minimization.

Findings

Achieves similar convergence rates as conventional models

Significantly reduces computational cost

Demonstrates effectiveness through numerical tests

Abstract

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.