Extended finite element method (XFEM) achieves unprecedented success in crack growth simulations, in particular in 2D. However, challenges still remain for a 3D crack growth simulation. Among other issues like increased computational expense, one challenge is robust and efficient geometry representation of the non-planar surface and the usually curved front of the 3D crack, two unique features distinguished from its 2D counterpart. With a special emphasis on accurate geometries of the 3D crack, we propose in this paper a B-spline ruled surface (BRS), a smooth, explicit, meshfree crack representation algorithm, which offers good geometrical accuracy, good numerical efficiency, and no mesh-resolution dependence. The technique accurately represents the geometry of an evolving 3D crack in a lightweight manner and leads to highly efficient point-surface projections for level set values. The technique is applied to modeling arbitrary 3D crack propagation in context of the Improved XFEM (IXFEM) that has been developed to eliminate the daunting issues of linear dependence and ill-conditioning of the standard XFEM (Tian and Wen (2015); Wen and Tian (2016)). Numerical tests show that IXFEM with the technique of crack description provides accurate solutions for the benchmarks of 3D crack propagation; the effectiveness of the proposed algorithm is demonstrated.

扩展有限元方法（XFEM）在裂纹扩展模拟中，尤其是在二维中，取得了空前的成功。但是，对于3D裂纹扩展模拟仍然存在挑战。除其他问题外，例如增加的计算费用，一个挑战是非平面表面和3D裂纹通常弯曲前部的鲁棒和有效的几何表示，这是区别于2D裂纹的两个独特特征。我们特别强调3D裂纹的精确几何形状，我们提出了一种B样条直纹曲面（BRS），一种平滑，显式，无网格的裂纹表示算法，该算法提供了良好的几何精度，良好的数值效率，并且没有网格分辨率依赖性。该技术以一种轻量级的方式准确地表示了一个不断发展的3D裂纹的几何形状，并为水平设置值提供了高效的点表面投影。该技术在改进的XFEM（IXFEM）的背景下被用于对任意3D裂纹扩展进行建模，该改进的XFEM消除了标准XFEM的线性依赖和不良条件这一艰巨的问题（Tian和Wen（2015）; Wen和Tian （2016））。数值测试表明，采用裂纹描述技术的IXFEM为3D裂纹扩展的基准提供了准确的解决方案。证明了所提算法的有效性。该技术在改进的XFEM（IXFEM）的背景下被用于对任意3D裂纹扩展进行建模，该改进的XFEM消除了标准XFEM的线性依赖和不良条件这一艰巨的问题（Tian和Wen（2015）; Wen和Tian （2016））。数值测试表明，采用裂纹描述技术的IXFEM为3D裂纹扩展的基准提供了准确的解决方案。证明了所提算法的有效性。该技术在改进的XFEM（IXFEM）的背景下被用于对任意3D裂纹扩展进行建模，该改进的XFEM消除了标准XFEM的线性依赖和不良条件这一艰巨的问题（Tian和Wen（2015）; Wen和Tian （2016））。数值测试表明，采用裂纹描述技术的IXFEM为3D裂纹扩展的基准提供了准确的解决方案。证明了所提算法的有效性。