Extended isogeometric analysis: a two-scale coupling FEM/IGA for 2D elastic fracture problems

Some of the key features of the isogeometric analysis, IGA, are the capacity of exactly representing the problem geometry, the use of the same basis functions to describe the geometry and the solution field, and a straightforward and automatic discretization refining scheme. The higher order continuity of the isogeometric approximation, important to correctly represent the domain geometry, can be a problem to approximate the displacement field in the neighbourhood of a crack. The eXtended Isogeometric Analysis (XIGA) overcomes this obstacle, enlarging the approximate space of IGA. This is achieved by incorporating customized functions, using the enrichment strategy of the Generalized/eXtended Finite Element Method. When these functions are unknown, they can be computed from the solution of local boundary value problems embracing the crack, and a global–local iterative procedure is established. Here this procedure is firstly proposed to combine FEM and isogeometric approximations, denoted XIGA gl . The effectiveness of this approach is investigated in terms of convergence rates and numerical stability. The method is applied to two-dimensional fracture mechanics problems. The numerical experiments show the importance of using the isogeometric approximation to recover more accurate solutions and minimize the deterioration of the conditioning of the related stiffness matrix.