科研之友
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摘要
Owing to the solution space enriched with a priori known knowledge through the partition of unity, the extended or generalized finite element method (XFEM/GFEM) has been widely applied to the modeling of localized failure in solids. As far as cohesive cracks are concerned in this work, on the one hand, the XFEM enriched with Heaviside or the shifted one with respect to nodal values) is of optimal accuracy, but is prone to the issue of ill-conditioned or even singular system matrix for arbitrary crack propagation. On the other hand, the stable XFEM (s-XFEM) circumvents the ill-conditioning issues by introducing the residual Heaviside function with respect to its linear interpolant, at the cost of losing accuracy (e.g., spurious stress locking) for discontinuities with non-uniform displacement jumps. Aiming to reconcile the incompatibility between accuracy of the solution and conditioning of the system matrix, this work addresses a simple yet effective improved stable XFEM (Is-XFEM) with a novel enrichment function. Both the XFEM and s-XFEM are recovered as its particular cases, with different approximation schemes for the bridging scale which incorporates information from both the continuous coarse scale and the discontinuous fine scale. In the proposed Is-XFEM, as the enrichment basis functions are not contained in the standard polynomial space, conditioning of the resulting system matrix is nearly insensitive to the mesh/discontinuity configuration even if the crack path gets close to element nodes. Furthermore, the Is-XFEM with a small stabilization parameter is almost of the same accuracy as the XFEM. Particularly, pathological numerical results polluted by spurious stress locking exhibited in the s-XFEM are not observed across a fully softened discontinuity with non-uniform displacement jumps. As only the enrichment function is modified, the proposed Is-XFEM, together with appropriate crack propagation criterion and tracking algorithm, can be implemented easily in the existing XFEM codes. Several representative numerical simulations of element and structure benchmark tests are presented to validate the proposed Is-XFEM, regarding both accuracy of the solution and conditioning of the system matrix.
