Phase-field models for brittle fracture in anisotropic materials result in a fourth-order partial differential equation for the damage evolution. This necessitates a $$\mathcal {C}^1$$ continuity of the basis functions. Here, Powell-Sabin B-splines, which are based on triangles, are used for the approximation of the field variables as well as for the the description of the geometry. The use of triangles makes adaptive mesh refinement and discrete crack insertion straightforward. Bezier extraction is used to cast the B-splines in a standard finite element format. A procedure to impose Dirichlet boundary condition is provided for these elements. The versatility and accuracy of the approach are assessed in two case studies, featuring crack kinking and zig-zag crack propagation. It is also shown that the adaptive refinement well captures the evolution of the phase field.

中文翻译：

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Powell-Sabin B-Splines 在裂纹扭结的高阶相场模型中的使用

各向异性材料脆性断裂的相场模型产生了损伤演化的四阶偏微分方程。这需要基函数的 $$\mathcal {C}^1$$ 连续性。此处，基于三角形的 Powell-Sabin B 样条用于场变量的近似以及几何形状的描述。三角形的使用使自适应网格细化和离散裂纹插入变得简单明了。Bezier 提取用于以标准有限元格式转换 B 样条。为这些元素提供了施加 Dirichlet 边界条件的程序。该方法的多功能性和准确性在两个案例研究中得到评估，以裂纹扭结和锯齿形裂纹扩展为特征。