Stability of incompressible formulations enriched with X-FEM

The treatment of (near-)incompressibility is a major concern for applications involving rubber-like materials, or when important plastic flows occurs as in forming processes. The use of mixed finite element methods is known to prevent the locking of the finite element approximation in the incompressible limit. However, it also introduces a critical condition for the stability of the formulation, called the inf–sup or LBB condition. Recently, the finite element method has evolved with the introduction of the partition of unity method. The eXtended Finite Element Method (X-FEM) uses the partition of unity method to remove the need to mesh physical surfaces or to re-mesh them as they evolve. The enrichment of the displacement field makes it possible to treat surfaces of discontinuity inside finite elements. In this paper, some strategies are proposed for the enrichment of low order mixed finite element approximations in the incompressible setting. The case of holes, material interfaces and cracks are considered. Numerical examples show that for well chosen enrichment strategies, the finite element convergence rate is preserved and the inf–sup condition is passed.