Optimally convergent high-order X-FEM for problems with voids and inclusions

Abstract

Solution of multiphase problems shows discontinuities across the material interfaces, which are usually weak. Using the eXtended Finite Element Method (X-FEM), these problems can be solved even for meshes that do not match the geometry. The basic idea is to enrich the interpolation space by means of a ridge function that is able to reproduce the discontinuity inside the elements. This approach yields excellent results for linear elements, but fails to be optimal if high-order interpolations are used. In this work, we propose a formulation that ensures optimal convergence rates for bimaterial problems. The key idea is to enrich the interpolation using a Heaviside function that allows the solution to represent polynomials on both sides of the interface and, provided the interface is accurately approximated, it yields optimal convergence rates. Although the interpolation is discontinuous, the desired continuity of the solution is imposed modifying the weak form. Moreover, in order to ensure optimal convergence, an accurate description of the interface (which also defines an integration rule for the elements cut by the interface) is needed. Here, we comment on different options that have been successfully used to integrate high-order X-FEM elements, and describe a general algorithm based on approximating the interface by piecewise polynomials of the same degree that the interpolation functions.