A stabilization technique for the regularization of nearly singular extended finite elements

An improved stable XFEM (Is-XFEM) with a novel enrichment function for the computational modeling of cohesive cracks

“…Lang et al [41] proposed using a linear geometric preconditioner based on the nodal basis functions and removing those dofs associated to nodes with small supports. Loehnert [37] suggested a stabilization technique in which zero eigenmodes are filtered out from the stiffness matrix of elements with enriched nodes. However, most of these preconditioners/stabilization have to be built in each iteration for nonlinear problems, bringing about burdensome computational overheads.…”

“…To remedy the resulting ill-conditioning issue, various approaches have been suggested in the literature. A natural scheme is to modify the unfavorable mesh with a small intersection ratio, either by moving the crack onto (or away from) the element node [35,36], or vice versa, by moving the node onto (or away from) the discontinuity; see [37] for a brief review. However, these mesh adapting or updating strategies typically encounters issues of efficiency and robustness for problems involving complex geometries and moving interfaces.…”

A regularization technique for the XFEM: extension to finite deformations, inelastic material behaviour and multifield problems

“…by means of a Gram-Schmidt projection and a subsequent reorthogonalization. In [1] a very similar procedure is performed for the zero eigenspace of a symmetric element stiffness matrix. The remaining zero singular subspaceÛ = U 0 \W 0 andV = V 0 \W 0 is only non-empty if linear dependencies among standard and enriched degrees of freedom or among enriched degrees of freedom exist that should not be present.…”

“…In the most general case the element stiffness matrix K e resulting from a consistent linearization of the weak form can become non-symmetric. For this case the eigenvalue decomposition that is performed for all elements that possess only enriched nodes in [1] is replaced by a singular value decomposition (SVD) to avoid complex numbers in the analysis. Here, the singular values Σ and the corresponding singular vectors contained in U and V are split into the non-zero singular space (Ū andV ) with the singular valuesΣ and the zero singular space with the singular vectors U 0 and V 0 .…”

“…However, this technique also has a few drawbacks as reported in literature. Here, a generalization of the stabilization technique originally presented in [1] extended to geometrically and materially non-linear problems is proposed which does not show any of the drawbacks alternative methods exhibit.…”