Strong imposing Dirichlet boundary conditions remain a challenge for eXtended Finite Element Methods (XFEMs) or eXtended Isogeometric Analysis (XIGA) of discontinuous problems. Moreover, the physical meaning of the additional Degrees of Freedom (DOFs) in the displacement expression of XFEMs or XIGA is not clarified. To address these issues, we proposed a new method that combines XIGA and B++ splines for modeling the fracture behaviors of single and multiple cracks in 2D elasticity solids. We adopt the overlapping trimming curves to represent the crack curves. The control points that the basis functions are non-zeros for the penetrating elements in XIGA are changed into the double-layer collocation points on the crack curves in our proposed method. In doing so, the DOFs of control points of the elements penetrated by the crack in the XIGA are replaced by those of the double-layer collocation points on the crack curves. The movement of the crack edges is simulated by the displacement of collocation points. Thus, the physical meaning of the DOFs of the elements penetrated by the cracks is clarified. As the basis functions of the collocation points on the crack boundary satisfy the Kronecker delta property, the presented method allows strong imposing Dirichlet boundary conditions on the crack edges. The stress intensity factors (SIFs) are calculated by adopting the interaction integral technology. Numerical examples verify the accuracy of the proposed method.

强势的Dirichlet边界条件仍然是不连续问题的扩展有限元方法（XFEM）或扩展等几何分析（XIGA）的挑战。此外，XFEM或XIGA的位移表达中的附加自由度（DOF）的物理含义尚不清楚。为了解决这些问题，我们提出了一种结合XIGA和B ++样条曲线的新方法来对2D弹性实体中单个和多个裂纹的断裂行为进行建模。我们采用重叠的修整曲线来表示裂纹曲线。在我们提出的方法中，将XIGA中穿透元素的基函数为非零的控制点更改为裂纹曲线上的双层配置点。这样做时，在XIGA中被裂纹穿透的元素的控制点的自由度被裂纹曲线上的双层配置点所取代。裂纹边缘的移动是通过并置点的位移来模拟的。因此，阐明了被裂纹穿透的元素的自由度的物理含义。裂纹边界上的并置点的基函数满足Kronecker增量的性质，提出的方法允许在裂纹边缘强加Dirichlet边界条件。应力强度因子（SIFs）是通过采用交互积分技术来计算的。数值算例验证了所提方法的准确性。