A well‐conditioned and optimally convergent XFEM for 3D linear elastic fracture

Abstract

A variation of the extended finite element method for 3D fracture mechanics is proposed. It utilizes global enrichment and point-wise as well as integral matching of displacements of the standard and enriched elements in order to achieve higher accuracy, optimal convergence rates and improved conditioning for two and three Preprint submitted to Elsevier Science 29 January 2015

Minimum energy multiple crack propagation. Part-II: Discrete solution with XFEM

“…Here, the energy of the system (1) is minimised for a given (fixed) total fracture length increment. This approach is suitable for both stable (5) and unstable (6) crack growth regimes; however, the fracture front is required to be stable (2). In other words, the energy function (1) needs to be convex at least within the constrained solution space.…”

“…Here, the external load magnitude is incremented which is then followed by the extension of the crack tips that have supercritical crack tip energy release rates. The method is limited to stable fracture growth (5) and a stable fracture front (2). In other words, the energy function (1) needs to be convex.…”

“…This helps resolve the crack tip field with better accuracy than with topological enrichment [22], which usually means enriching the nodes of a single element at the crack tip. On the other hand, the present strategy still preserves a relatively well-conditioned system of equations [2,1,17] in comparison to geometric enrichment [22], which typically refers to enriching the nodes of a patch of elements irrespectively of the underlying mesh.…”

Stable 3D XFEM/vector level sets for non‐planar 3D crack propagation and comparison of enrichment schemes

“…Although the inclusion of singularities in the FE approximation provides significant benefits by increasing the accuracy of the solution and minimizing the need for mesh refinement, it also leads to some implementation problems. Those problems are mostly related to numerical integration where special techniques have been proposed for 2D and 3D elements, conditioning of the resulting system matrices, which can be dealt with through the use of special preconditioners, degree of freedom gathering or stabilization, and blending of the enriched with the standard part of the approximation for which several remedies have been proposed such as weight function blending and enhanced/assumed strain elements . Other integration techniques were proposed to simplify the integration of functions with strong or weak discontinuities and singularities some of which do not require element subdivision …”

Minimum energy multiple crack propagation. Part III: XFEM computer implementation and applications

“…1 In XFEM, the classic finite element approximation is enriched with additional functions that can capture the essential features of discontinuities, high gradients and singularities associated with cracks in LEFM. In turn, XFEM can offer accurate and optimally convergent numerical solution for the elastic field [25,22,23,2,1]. The reader is referred to Part-II of our three-part paper for details on the XFEM approximation we use.…”