Mesh based construction of flat-top partition of unity functions

Effects of the smoothness of partitions of unity on the quality of representation of singular enrichments for GFEM/XFEM stress approximations around brittle cracks

“…The flat-top property of C k -GFEM PoU has shown to be a very good alternative for incorporating the information provided by the singular enrichment functions (6) of conventional XFEM [10] in the ansatz. The flat-top allows the reproduction of the singular enrichment with greater fidelity, and this property has been noted as a desirable feature of a PoU function [47,48]. This effect can be noted from the low dependence on the pattern of enrichment, and the conclusion is supported by the higher eigenvalues of the stiffness matrix of C k -GFEM, compared to the standard C 0 version (continuous lines in Fig.…”

“…Some advantage in terms of good reproduction of the enrichments of the flat-top-featured PoU functions has also been reported in [48,45].…”

“…Each curve corresponds to a specific degree b of uniform polynomial reproducibility. In other words, quasi-singular stiffness matrices can be obtained with C k -GFEM when higher-degree polynomial enrichments, with the same nature as in (5), are uniformly applied on the domain, even though linear dependence theoretically does not occur when the PoU functions are not polynomial [24,46,45,48,75]. However, such intense conditioning deterioration related to the polynomial enrichment does not occur when the enrichment is localized, as in adaptive procedures [2,3].…”

“…A more recent approach that produces a PoU with an a priori defined flat-top dimension is the work of [48]. This methodology considers quadrilateral finite element meshes, and the topology of the flat-top portions is a consequence of the mesh because the transitions, between PoU functions, are defined by narrow bands along the edges, whose width is previously chosen.…”

Local refinement of flat‐top partition of unity based high‐order approximation

“…However, the most commonly used method to avoid the linear dependence is to add flat‐top re gions into the original PU‐based mesh system. () He et al and An et al has proven that it is linearly independent for the regularly patterned flat‐top PU mesh in one‐ and two‐dimensional spaces.…”