Large sliding contact along branched discontinuities with X-FEM

On the construction of approximation space to model discontinuities and cracks with linear and quadratic extended finite elements

“…In the literature, the answer is far from obvious. Authors suggested that there is indeed a sensitivity of the solution with respect to the position of the interface; nonetheless, they considered this sensitivity as a numerical side-effect and complex numerical solutions have been elaborated to deal with this issue [21,24]. Although those techniques may work with linear elements, very little is said about their efficiency with quadratic elements.…”

“…Those numerical issues have been studied in the case of X-FEM formulations [24]. With X-FEM, the enrichment functions and the classical shape functions are almost collinear when the interface cuts the mesh near one of its nodes (Fig.…”

“…The speed of increase of the condition number is related to the distance between the interface and the nodes of the mesh. The condition number soars when either the distance N 1 IP goes to zero or when N 2 IP goes to zero First of all, we can consider the elimination of degrees of freedom, based on straightforward criteria [18,24]. Those criteria are rough estimates of the "Heaviside information" on both sides of the discontinuity.…”

“…The criterion used here replaces the measurement of volumes [18,24] by the measurement of distances between the nodes of the mesh and the interface on cut edges. When these criteria are associated to a relocation of the interface they are usually named fit-to- .…”

“…Those concerns have been described in [17,[21][22][23] but have raised little interest since recent publications [24,25]. Firstly, they are related to the ill conditioning of strong or weak discontinuous approximations in the general case of non-conforming interfaces and secondly, to the numerical issues related to "geometrical enrichment" techniques, near the crack-tip.…”