A new extended isogeometric boundary element method (XIGABEM) formulation is proposed for simulating multiple fatigue crack propagation in two-dimensional domains. The classical use of NURBS in isogeometric formulations is further extended by repeated knot insertion to introduce a $C^{-1}$ continuity within the approximation space as an elegant approach to representing geometrical discontinuities at crack intersections. This strategy is also used to restrict the enrichment term to the portion of the NURBS defining the tip, where it is necessary. At this near-tip zone, the linear elastic fracture mechanics solutions are embedded into the displacement approximation to represent the theoretical square root behaviour. The enrichment procedure introduces just two degrees of freedom per crack tip, and a tying constraint is used to yield a square linear system. In this direct approach, the stress intensity factors (SIFs) are found as terms in the solution vector without requiring post-processing techniques.

Several examples are presented to illustrate the application of the XIGABEM. The accuracy of the results compares favourably against those from the literature, and also against solutions obtained from unenriched and enriched indirect methods that employ the J-integral for SIF extraction. Furthermore, the proposed direct approach is capable of significantly reducing the execution time.

提出了一种新的扩展等几何边界元方法 (XIGABEM) 公式，用于模拟二维域中的多个疲劳裂纹扩展。NURBS 在等几何公式中的经典应用通过重复插入结点进一步扩展，以引入$C^{-1}$近似空间内的连续性作为表示裂纹交叉处几何不连续性的一种优雅方法。此策略还用于将富集项限制在 NURBS 中定义尖端的部分，这是必要的。在这个近尖端区域，线弹性断裂力学解被嵌入到位移近似中以表示理论平方根行为。富集过程只为每个裂纹尖端引入两个自由度，并使用绑定约束来生成方形线性系统。在这种直接方法中，应力强度因子 (SIF) 作为解向量中的项被发现，无需后处理技术。

给出了几个例子来说明 XIGABEM 的应用。结果的准确性优于文献中的结果，也与使用 J 积分进行 SIF 提取的未富集和富集间接方法获得的解决方案相比。此外，所提出的直接方法能够显着减少执行时间。