Extended isogeometric analysis using B++ splines for strong discontinuous problems

Abstract

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.