Stabilization of the extended finite element method for stiff embedded interfaces and inclusions

This article investigates the performance of the XFEM for stiff embedded interfaces and inclusions. It is known that the XFEM may lead to ill-conditioned stiffness matrices and oscillations in the interface traction field, the severity of which depends on the underlying basis functions, as well as on the orientation of the interface. The jumps at the discontinuity are shown to contain quadratic bubble residuals that can introduce oscillatory behavior. Those residuals are worsened by the ill-conditioning of the system with very stiff interfaces. A variationally consistent method is proposed to overcome the oscillatory behavior and ill-conditioning, in which the assumed strain method is developed to directly eliminate bubble residuals and deploy Legendre polynomials and explore their orthogonality properties to improve the conditioning of the stiffness matrices. Numerical examples illustrate the efficiency and generality of the proposed approach at both element and structural levels. The new approach is shown to be robust for imposing Dirichlet type boundary conditions at the crack interface, such as crack closure and initially rigid cohesive laws. The effects of numerical oscillations on the prediction of effective properties of composites in XFEM-based computational homogenization procedures are also discussed.

Automated summary

This article investigates the performance of the XFEM for stiff embedded interfaces and inclusions, proposing a variational consistent method to overcome oscillatory behavior and ill-conditioning. The new approach is shown to be efficient and general at both element and structural levels.