The surface energy a phase‐field approach to brittle fracture in anisotropic materials is also anisotropic and gives rise to second‐order gradients in the phase field entering the energy functional. This necessitates C¹ continuity of the basis functions which are used to interpolate the phase field. The basis functions which are employed in isogeometric analysis (IGA), such as nonuniform rational B‐splines and T‐splines naturally possess a higher order continuity and are therefore ideally suited for phase‐field models which are equipped with an anisotropic surface energy. Moreover, the high accuracy of spline discretizations, also relative to their computational demand, significantly reduces the fineness of the required discretization. This holds a fortiori if adaptivity is included. Herein, we present two adaptive refinement schemes in IGA, namely, adaptive local refinement and adaptive hierarchical refinement, for phase‐field simulations of anisotropic brittle fracture. The refinement is carried out using a subdivision operator and exploits the Bézier extraction operator. Illustrative examples are included, which show that the method can simulate highly complex crack patterns such as zigzag crack propagation. An excellent agreement is obtained between the solutions from global refinement and adaptive refinement, with a reasonable reduction of the computational effort when using adaptivity.

中文翻译：

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各向异性脆性断裂相场建模的自适应等几何分析

各向异性材料中脆性断裂的相场方法的表面能也是各向异性的，并在进入能量函数的相场中产生二阶梯度。这需要C ¹用于内插相场的基函数的连续性。等几何分析（IGA）中使用的基础函数，例如非均匀有理B样条和T样条，自然具有较高的连续性，因此非常适合配备各向异性表面能的相场模型。此外，样条离散化的高精度（也相对于它们的计算需求）显着降低了所需离散化的精细度。如果包括适应性，那么这很重要。在此，我们针对各向异性脆性断裂的相场模拟，提出了IGA中的两种自适应细化方案，即自适应局部细化和自适应分层细化。使用细分运算符进行细化并利用Bézier提取运算符。包括说明性示例，这些示例表明该方法可以模拟高度复杂的裂纹模式，例如之字形裂纹扩展。全局细化和自适应细化的解决方案之间获得了极好的协议，并且在使用自适应性时合理地减少了计算工作量。