A locking-free and optimally convergent discontinuous-Galerkin-based extended finite element method for cracked nearly incompressible solids

The extended finite element method (XFEM) is an efficient way to include discontinuities, such as a crack, into a finite element mesh. The singularity at the crack tip restricts standard finite element methods to converge with a rate of at most 1/2 for the stresses, and 1 for the displacements, with respect to the mesh size. This is true for cracks in incompressible materials as well, when any of the standard techniques to sidestep locking is adopted. To attain an optimal convergence rate of 1 for stresses and of 2 for displacements with piecewise affine elements, it is necessary to enrich the finite element space with singular basis functions. The support of these singular functions is the entire plane, but to avoid decreasing the sparsity of the stiffness matrix too much, each of them is then generally localized to a neighborhood of the crack tip by multiplying by a cutoff function or a subset of a partition-of-unity basis. For nearly incompressible materials, however, the resulting basis functions no longer contain incompressible displacement fields, and hence they either lead to locking or suboptimal convergence rates. To overcome this problem, we introduce here an XFEM with optimal convergence rate and without the problem of locking for nearly incompressible materials, i.e., it possesses an error bound that does not diverge as Poisson’s ratio approaches 0.5. The method is based on a primal, or one-field formulation of a discontinuous Galerkin method that we introduced earlier. This one-field formulation is obtained through the introduction of a lifting operator, but unlike most lifting operators which map inter-element discontinuities into elementwise polynomials, ours maps such discontinuities into spaces enriched with the singular behavior of the solution. This is the key idea for the method to be simultaneously locking-free and optimally convergent.