Adaptive mesh refinement for the phase field method: A FEniCS implementation

Abstract

This paper reports insightful implementation details of the global adaptive refinement procedure for the phase-field method recently published in Freddi and Mingazzi (2022). Phase field approaches reproduce cracks within solids in a smeared manner. The small transition zone between broken and unbroken material, whose width is controlled by an internal scale length parameter, permits to precisely replicate complex sharp crack topologies only if an extremely fine mesh is adopted. Starting from a coarse mesh, the proposed refinement process utilizes an energetic criterion to selectively refine the elements on which cracks may propagate. In fully broken areas, where the phase field is no longer evolving, a specific refinement is adopted to capture the high displacement gradient. The implementation is performed within the open-source finite element software FEniCS (ver. 19.1.0) which provides a framework for automated solutions of partial differential equations. The fundamental aspects of the code are described starting from the functional definition to the various steps of the refinement technique. A representative example is illustrated to supply further information on the code functionality. The code can be downloaded from https://github.com/LorenzoMingazzi/AGu-AGal and be used to easily apply the proposed refinement strategy to different problems or as a starting point for more sophisticated formulations.