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Effects of the smoothness of partitions of unity on the quality of representation of singular enrichments for GFEM/XFEM stress approximations around brittle cracks. (English) Zbl 1423.74849
Summary: The convergence rates of the conventional generalized/extended finite element method (GFEM/XFEM) in crack modeling are similar to the convergence rates of the finite element method (FEM) [P. Laborde et al., Int. J. Numer. Methods Eng. 64, No. 3, 354–381 (2005; Zbl 1181.74136); E. Béchet et al., Int. J. Numer. Methods Eng. 64, No. 8, 1033–1056 (2005; Zbl 1122.74499)] unless the crack tip enrichment functions are applied in a subdomain with fixed dimension, independently of the mesh parameter \(h\) and also demanding some special care for blending along the transition zone [E. Chahine et al., C. R., Math., Acad. Sci. Paris 342, No. 7, 527–532 (2006; Zbl 1276.74035); J. E. Tarancón et al., Int. J. Numer. Methods Eng. 77, No. 1, 126–148 (2009; Zbl 1195.74199)]. Thus, to improve convergence rates, more degrees of freedom (DOF) are generated due to the larger quantity of enriched nodes. This work seeks to identify and understand the advantages of better capturing the information provided by singular enrichments over mesh-based smooth partitions of unity (PoU). Such PoU with higher regularity can be built through the so-called \(C^k\)-GFEM framework, following C. A. Duarte et al. [Comput. Methods Appl. Mech. Eng. 196, No. 1–3, 33–56 (2006; Zbl 1120.74816)], based on a moving least square of degree zero and considering mesh-based smooth weighting functions associated with arbitrary polygonal clouds. The purpose herein is to investigate some possible advantages of mesh-based smooth PoU for modeling discontinuities and singularities, in two-dimensional problems of linear elastic fracture mechanics, in such a fashion that the discretization error associated to stress discontinuities inherent in standard \(C^0\)-continuous GFEM/XFEM approximations is eliminated. The procedure shares features similar to the standard FEM regarding domain partition and numerical integration but, as neither the PoU nor the enrichment functions are defined in natural domains, the integrations are performed using only global coordinates. The approximation capabilities of \(C^k\)-GFEM discretizations with different patterns of singular enrichment distribution are investigated by analyzing the convergence rates of the \(h\) and \(p\) versions, considering global measures in terms of strain energy and \(\mathfrak{L}^2\)-norm of displacements. The effects on stability are also verified by analyzing the evolution of the condition number. The effect of smoothness on conditioning is investigated and the eigenvalues distributions are used to identify the several aspects involved: the smoothness of the PoU, the different types of enrichment functions and the pattern of enrichment. The performance of the smooth approximations is compared to the \(C^0\) counterparts built using conventional \(C^0\) FEM-based PoU. It is shown that smoothness in the presence of extrinsically applied singular enrichments is important, as the information provided by such enrichment is better captured. In addition, there are no stress jumps around singularities, reducing error propagation beyond the neighborhood of the singularity.

MSC:
74R10 Brittle fracture
74S05 Finite element methods applied to problems in solid mechanics
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
Software:
XFEM; Mfree2D
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