The phase field model is a widely used mathematical approach for describing crack propagation in continuum damage fractures. In the context of phase field fracture simulations, adaptive finite element methods (AFEM) are often employed to address the mesh size dependency of the model. However, existing AFEM approaches for this application frequently rely on heuristic adjustments and empirical parameters for mesh refinement. In this paper, we introduce an adaptive finite element method based on a recovery type posteriori error estimates approach grounded in theoretical analysis. This method transforms the gradient of the numerical solution into a smoother function space, using the difference between the recovered gradient and the original numerical gradient as an error indicator for adaptive mesh refinement. This enables the automatic capture of crack propagation directions without the need for empirical parameters. We have implemented this adaptive method for the Hybrid formulation of the phase field model using the open-source software package FEALPy. The accuracy and efficiency of the proposed approach are demonstrated through simulations of classical 2D and 3D brittle fracture examples, validating the robustness and effectiveness of our implementation.

翻译：相场模型是描述连续介质损伤断裂中裂纹扩展的常用数学方法。在相场断裂模拟中，常采用自适应有限元方法（AFEM）以解决模型的网格尺寸依赖性问题。然而，现有针对该应用的自适应有限元方法通常依赖启发式调整和经验参数进行网格细化。本文提出一种基于理论分析的恢复型后验误差估计的自适应有限元方法。该方法将数值解的梯度转换至更光滑的函数空间，利用恢复梯度与原始数值梯度之间的差异作为自适应网格细化的误差指示器，从而无需经验参数即可自动捕捉裂纹扩展方向。我们已基于开源软件包FEALPy实现了该自适应方法在相场模型混合公式中的应用。通过经典二维与三维脆性断裂算例的模拟，验证了所提方法的精度与效率，证明了我们实现的鲁棒性与有效性。