Extended Powell–Sabin finite element scheme for linear elastic fracture mechanics

Abstract

Powell–Sabin B-splines, which are based on triangles, are employed in the framework of the extended finite element method (XFEM) for fracture analysis. This avoids the necessity of remeshing in discrete fracture models and increases the solution accuracy around the crack tip. Powell–Sabin B-splines are -continuous throughout the whole domain. The stresses around crack tips are captured more accurately than when using elements with a standard Lagrangian interpolation. Although Powell–Sabin B-splines do not hold the Kronecker-delta property, the Heaviside function and the tip enrichment function are confined to the cracked elements only, similar to the traditional XFEM but different from the extended isogeometric method. In addition, Powell–Sabin B-splines still hold -continuous throughout cracked elements. There is no need to lower the continuity at element boundaries, to confine basis function support in cracked elements. Shifting is used to ensure compatibility with the surrounding discretization. The sub-triangle technique is employed for the numerical integration over crack elements. The versatility and accuracy of the approach to simulate crack problems are assessed in case studies, featuring mode-I and mixed-mode crack problems.