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.

In recent years, topology optimization (TO) has gained widespread attention as a powerful structural design method. However, its application remains challenging due to the deep expertise and extensive development effort required. Traditional TO methods, tightly coupled with computational mechanics like finite element method (FEM), result in intrusive algorithms demanding a comprehensive system understanding. This paper presents SOPTX, a TO package based on FEALPy, which implements a modular architecture that decouples analysis from optimization, supports multiple computational backends (NumPy, PyTorch, JAX), and achieves a non-intrusive design paradigm. Core innovations include: (1) cross-platform design that supports multiple computational backends, enabling efficient algorithm execution on central processing units (CPUs) and flexible acceleration using graphics processing units (GPUs), while leveraging automatic differentiation (AD) technology for efficient sensitivity computation of objective and constraint functions; (2) fast matrix assembly techniques that overcome the performance bottlenecks of traditional numerical integration methods, significantly accelerating finite element computations and enhancing overall efficiency; (3) a modular framework supporting TO problems for arbitrary dimensions and meshes, allowing flexible configuration and extensibility of optimization workflows through a rich library of composable components. Using the density-based method for the classic compliance minimization problem with volume constraints as an example, numerical experiments demonstrate SOPTX's high efficiency in computational speed and memory usage, while showcasing its strong potential for research and engineering applications.