  The extended finite element method (XFEM) was introduced in 1999 to treat
problems involving discontinuities with no or minimal remeshing through
appropriate enrichment functions. This enables elements to be split by a
discontinuity, strong or weak and hence requires the integration of
discontinuous functions or functions with discontinuous derivatives over
elementary volumes. Moreover, in the case of open surfaces and singularities,
special, usually non-polynomial functions must also be integrated. A variety of
approaches have been proposed to facilitate these special types of numerical
integration, which have been shown to have a large impact on the accuracy and
the convergence of the numerical solution. The smoothed extended finite element
method (SmXFEM) [1], for example, makes numerical integration elegant and
simple by transforming volume integrals into surface integrals. However, it was
reported in [1, 2] that the strain smoothing is inaccurate when non-polynomial
functions are in the basis. This is due to the constant smoothing function used
over the smoothing domains which destroys the effect of the singularity. In
this paper, we investigate the benefits of a recently developed Linear
smoothing procedure [3] which provides better approximation to higher order
polynomial fields in the basis. Some benchmark problems in the context of
linear elastic fracture mechanics (LEFM) are solved to compare the standard
XFEM, the constant-smoothed XFEM (Sm-XFEM) and the linear-smoothed XFEM
(LSm-XFEM). We observe that the convergence rates of all three methods are the
same. The stress intensity factors (SIFs) computed through the proposed
LSm-XFEM are however more accurate than that obtained through Sm-XFEM. To
conclude, compared to the conventional XFEM, the same order of accuracy is
achieved at a relatively low computational effort.
